Exponents in Factor Form (A)

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Unit 6: Expressions with exponents

About this unit, exponent properties intro.

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Negative exponents

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Exponent properties (integer exponents)

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Scientific notation intro

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Interactive Math Resources for Classroom Discussion, Inquiry and Distance Learning

factoring exponents worksheet

Factoring Expressions with Exponents

Definition: To factor a polynomial is to write the addition of two or more terms as the product of two or more terms.

Consider the addition of the two numbers 24 + 30. Notice that they are both multiples of 6.

We could write

\begin{align*}&24 + 30 \\&= 6 \times 4 + 6 \times 5\\& = 6(4 + 5)\end{align*}

The factors are ‘6’ and ‘(4+5)’.

Factoring numbers in this way doesn’t mean too much, however changing the form from a sum (the addition of terms) to the product (the multiplication of terms) can be informative for polynomials. Polynomials are introduced on the next page .

What we wish to accomplish on this page is being able to factor an expression of several terms with one, two or possibly three variables.

15ab^4+20a^3b=5ab(3b^3+4a^2)

Learning method 1: Follow the Khan academy lessons

Greatest Common Factor of Monomials

Taking Common Factor from Binomial

Learning method 2: Open the worksheet below, watch the video, complete the worksheet, check the solutions.

Worksheet: Grade 10 FMP Factor Polynomials

Video: A few examples

Solutions:  10 FMP Factor Polynomial Solutions

Learning method 3: Follow the examples

14x^3+21x^2

Let’s list the factors of each term:

\[14x^3+21x^2=7\cdot 2 \cdot x \cdot x \cdot x+ 7 \cdot 3 \cdot x \cdot x\]

Let’s underline the factors that appear in both terms:

\[14x^3+21x^2=\underline{7}\cdot 2 \cdot \underline{x} \cdot \underline{x} \cdot x+ \underline{7} \cdot 3 \cdot \underline{x} \cdot \underline{x}\]

Let’s take out of the bracket what is common to both terms, and leave in the bracket what is left from each term:

\[14x^3+21x^2=7x^2(2x+3)\]

Notice that the highest common factor of 15, 20 is 5.

Following the same steps as for example 1 we have:

\begin{align*}&15ab^4+20a^3b\\[10pt]&=5\cdot 3\cdot a\cdot b \cdot b \cdot b \cdot b + 5 \cdot 4 \cdot a \cdot a \cdot a \cdot b \\[10pt]&=\underline{5}\cdot 3\cdot \underline{a}\cdot \underline{b} \cdot b \cdot b \cdot b + \underline{5} \cdot 4 \cdot \underline{a} \cdot a \cdot a \cdot \underline{b} \\[10pt]&=5ab(3b^3+4a^2)\end{align*}

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1.5: Factoring Polynomials

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Learning Objectives

In this section students will:

  • Factor the greatest common factor of a polynomial.
  • Factor a trinomial.
  • Factor by grouping.
  • Factor a perfect square trinomial.
  • Factor a difference of squares.
  • Factor the sum and difference of cubes.
  • Factor expressions using fractional or negative exponents.

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in Figure \(\PageIndex{1}\).

A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4.

The area of the entire region can be found using the formula for the area of a rectangle.

\[\begin{align*} A &= lw\\ &= 10x\times6x\\ &= 60x^2\; units^2 \end{align*}\]

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of \(A =s^2=4^2= 16\; units^2\). The other rectangular region has one side of length \(10x−8\) and one side of length \(4\), giving an area of

\[A =lw=4(10x−8)=40x−32\; \text{units}^2. \nonumber\]

So the region that must be subtracted has an area of

\[2(16)+40x−32= 40x\; \text{units}^2. \nonumber\]

The area of the region that requires grass seed is found by subtracting \(60x^2−40x\; \text{units}^2\). This area can also be expressed in factored form as \(20x (3x−2)\; \text{units}^2\). We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, \(4\) is the GCF of \(16\) and \(20\) because it is the largest number that divides evenly into both \(16\) and \(20\) The GCF of polynomials works the same way: \(4x\) is the GCF of \(16x\) and \(20x^2\) because it is the largest polynomial that divides evenly into both \(16x\) and \(20x^2\).

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Definition: Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

Howto: Given a polynomial expression, factor out the greatest common factor

  • Identify the GCF of the coefficients.
  • Identify the GCF of the variables.
  • Combine to find the GCF of the expression.
  • Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  • Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Example \(\PageIndex{1}\): Factoring the Greatest Common Factor

Factor \(6x^3y^3 +45x^2y^2+21xy\).

First, find the GCF of the expression. The GCF of \(6\), \(45\), and \(21\) is \(3\). The GCF of \(x^3\),\(x^2\), and \(x\) is \(x\). (Note that the GCF of a set of expressions in the form \(x^n\) will always be the exponent of lowest degree.) And the GCF of \(y^3\),\(y^2\), and \(y\) is \(y\). Combine these to find the GCF of the polynomial, \(3xy\).

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that

  • \(3xy(2x^2y^2)=6x^3y^3\),
  • \(3xy(15xy)=45x^2y^2\), and
  • \(3xy(7)=21xy\).

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

\[(3xy)(2x^2y^2+15xy+7) \nonumber\]

After factoring, we can check our work by multiplying. Use the distributive property to confirm that

\[(3xy)(2x^2y^2+15xy+7)=6x^3y^3+45x^2y^2+21xy \nonumber\]

Exercise \(\PageIndex{1}\)

Factor \(x(b^2−a)+6(b^2−a)\) by pulling out the GCF.

\((b^2−a)(x+6)\)

Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial \(x^2+5x+6\) has a GCF of \(1\), but it can be written as the product of the factors \((x+2)\) and \((x+3)\).

Trinomials of the form \(x^2+bx+c\) can be factored by finding two numbers with a product of \(c\) and a sum of \(b\). The trinomial \(x^2+10x+16\), for example, can be factored using the numbers \(2\) and \(8\) because the product of those numbers is \(16\) and their sum is \(10\). The trinomial can be rewritten as the product of \((x+2)\) and \((x+8)\).

FACTORING A TRINOMIAL WITH LEADING COEFFICIENT \(1\)

A trinomial of the form \(x^2+bx+c\) can be written in factored form as \((x+p)(x+q)\) where \(pq=c\) and \(p+q=b\).

Q&A: Can every trinomial be factored as a product of binomials?

No. Some polynomials cannot be factored. These polynomials are said to be prime .

Howto: Given a trinomial in the form \(x^2+bx+c\), factor it

  • List factors of \(c\).
  • Find \(p\) and \(q\), a pair of factors of \(c\) with a sum of \(b\).
  • Write the factored expression \((x+p)(x+q)\).

Example \(\PageIndex{2}\): Factoring a Trinomial with Leading Coefficient 1

Factor \(x^2+2x−15\).

We have a trinomial with leading coefficient \(1\), \(b=2\), and \(c=−15\). We need to find two numbers with a product of \(−15\) and a sum of \(2\). In Table \(\PageIndex{1}\), we list factors until we find a pair with the desired sum.

Now that we have identified \(p\) and \(q\) as \(−3\) and \(5\), write the factored form as \((x−3)(x+5)\).

We can check our work by multiplying. Use FOIL to confirm that \((x−3)(x+5)=x^2+2x−15\).

Q&A: Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Exercise \(\PageIndex{2}\)

Factor \(x^2−7x+6\).

\((x−6)(x−1)\)

Factoring by Grouping

Trinomials with leading coefficients other than \(1\) are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial \(2x^2+5x+3\) can be rewritten as \((2x+3)(x+1)\) using this process. We begin by rewriting the original expression as \(2x^2+2x+3x+3\) and then factor each portion of the expression to obtain \(2x(x+1)+3(x+1)\). We then pull out the GCF of \((x+1)\) to find the factored expression.

Factor by Grouping

To factor a trinomial in the form \(ax^2+bx+c\) by grouping, we find two numbers with a product of \(ac\) and a sum of \(b\). We use these numbers to divide the \(x\) term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

Howto: Given a trinomial in the form \(ax^2+bx+c\), factor by grouping.

  • List factors of \(ac\).
  • Find \(p\) and \(q\), a pair of factors of \(ac\) with a sum of \(b\).
  • Rewrite the original expression as \(ax^2+px+qx+c\).
  • Pull out the GCF of \(ax^2+px\).
  • Pull out the GCF of \(qx+c\).
  • Factor out the GCF of the expression.

Example \(\PageIndex{3}\): Factoring a Trinomial by Grouping

Factor \(5x^2+7x−6\) by grouping.

We have a trinomial with \(a=5\),\(b=7\), and \(c=−6\). First, determine \(ac=−30\). We need to find two numbers with a product of \(−30\) and a sum of \(7\). In the table below, we list factors until we find a pair with the desired sum.

So \(p=−3\) and \(q=10\).

\(5x^2−3x+10x−6\) Rewrite the original expression as \(ax^2+px+qx+c\).

\(x(5x−3)+2(5x−3)\) Factor out the GCF of each part

\((5x−3)(x+2)\) Factor out the GCF of the expression.

We can check our work by multiplying. Use FOIL to confirm that \((5x−3)(x+2)=5x^2+7x−6\).

Exercise \(\PageIndex{3}\)

  • \(2x^2+9x+9\)
  • \(6x^2+x−1\)

\((2x+3)(x+3)\)

\((3x-1)(2x+1)\)

Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

\[a^2+2ab+b^2={(a+b)}^2\]

\[a^2-2ab+b^2={(a-b)}^2\]

We can use this equation to factor any perfect square trinomial.

Perfect Square Trinomials

A perfect square trinomial can be written as the square of a binomial:

\[a^2+2ab+b^2=(a+b)^2\]

Howto: Given a perfect square trinomial, factor it into the square of a binomial

  • Confirm that the first and last term are perfect squares.
  • Confirm that the middle term is twice the product of \(ab\).
  • Write the factored form as \({(a+b)}^2\).

Example \(\PageIndex{4}\): Factoring a Perfect Square Trinomial

Factor \(25x^2+20x+4\).

Notice that \(25x^2\) and \(4\) are perfect squares because \(25x^2={(5x)}^2\) and \(4=2^2\). Then check to see if the middle term is twice the product of \(5x\) and \(2\). The middle term is, indeed, twice the product: \(2(5x)(2)=20x\). Therefore, the trinomial is a perfect square trinomial and can be written as \({(5x+2)}^2\).

Exercise \(\PageIndex{4}\)

Factor \(49x^2−14x+1\).

\({(7x−1)}^2\)

Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

\[a^2−b^2=(a+b)(a−b)\]

We can use this equation to factor any differences of squares.

Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

Howto: Given a difference of squares, factor it into binomials

  • Write the factored form as \((a+b)(a−b)\).

Example \(\PageIndex{5}\): Factoring a Difference of Squares

Factor \(9x^2−25\).

Notice that \(9x^2\) and \(25\) are perfect squares because \(9x^2={(3x)}^2\) and \(25=5^2\). The polynomial represents a difference of squares and can be rewritten as \((3x+5)(3x−5)\).

Exercise \(\PageIndex{5}\)

Factor \(81y^2−100\).

\((9y+10)(9y−10)\)

Q&A: Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

\[a^3+b^3=(a+b)(a^2−ab+b^2)\]

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

\[a^3−b^3=(a−b)(a^2+ab+b^2)\]

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.

\[x^3−2^3=(x−2)(x^2+2x+4)\]

The sign of the first 2 is the same as the sign between \(x^3−2^3\). The sign of the \(2x\) term is opposite the sign between \(x^3−2^3\). And the sign of the last term, \(4\), is always positive.

Sum and Difference of Cubes

We can factor the sum of two cubes as

We can factor the difference of two cubes as

Howto: Given a sum of cubes or difference of cubes, factor it

  • Confirm that the first and last term are cubes, \(a^3+b^3\) or \(a^3−b^3\).
  • For a sum of cubes, write the factored form as \((a+b)(a^2−ab+b^2)\). For a difference of cubes, write the factored form as \((a−b)(a^2+ab+b^2)\).

Example \(\PageIndex{6}\): Factoring a Sum of Cubes

Factor \(x^3+512\).

Notice that \(x^3\) and \(512\) are cubes because \(8^3=512\). Rewrite the sum of cubes as \((x+8)(x^2−8x+64)\).

After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check.

Exercise \(\PageIndex{6}\)

Factor the sum of cubes: \(216a^3+b^3\).

\((6a+b)(36a^2−6ab+b^2)\)

Example \(\PageIndex{7}\): Factoring a Difference of Cubes

Factor \(8x^3−125\).

Notice that \(8x^3\) and \(125\) are cubes because \(8x^3={(2x)}^3\) and \(125=5^3\). Write the difference of cubes as \((2x−5)(4x^2+10x+25)\).

Just as with the sum of cubes, we will not be able to further factor the trinomial portion.

Exercise \(\PageIndex{7}\)

Factor the difference of cubes: \(1000x^3−1\)

\((10x−1)(100x^2+10x+1)\)

Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, \(2x^{\tfrac{1}{4}}+5x^{\tfrac{3}{4}}\) can be factored by pulling out \(x^{\tfrac{1}{4}}\) and being rewritten as \(x^{\tfrac{1}{4}}(2+5x^{\tfrac{1}{2}})\).

Example \(\PageIndex{8}\): Factoring an Expression with Fractional or Negative Exponents

Factor \(3x{(x+2)}^{-\tfrac{1}{3}}+4{(x+2)}^{\tfrac{2}{3}}\).

Factor out the term with the lowest value of the exponent. In this case, that would be \({(x+2)}^{-\tfrac{1}{3}}\).

\[\begin{align*} &(x+2)^{-\tfrac{1}{3}}(3x+4(x+2))\qquad \text{Factor out the GCF }\\ &(x+2)^{-\tfrac{1}{3}}(3x+4x+8)\qquad \text{Simplify } \\ &(x+2)^{-\tfrac{1}{3}}(7x+8) \end{align*}\]

Exercise \(\PageIndex{8}\)

Factor \(2{(5a−1)}^{\tfrac{3}{4}}+7a{(5a−1)}^{−\tfrac{1}{4}}\).

\({(5a−1)}^{−\tfrac{1}{4}}(17a−2)\)

Access these online resources for additional instruction and practice with factoring polynomials.

1. Identify GCF

2. Factor Trinomials when a Equals 1

3. Factor Trinomials when a is not equal to 1

4. Factor Sum or Difference of Cubes

Key Equations

  • The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See Example .
  • Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See Example .
  • Trinomials can be factored using a process called factoring by grouping. See Example .
  • Perfect square trinomials and the difference of squares are special products and can be factored using equations. See Example and Example .
  • The sum of cubes and the difference of cubes can be factored using equations. See Example and Example .
  • Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example .

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  • Equation: 36r^4 + 29r^2 - 20 Show answer | Show step-by-step Answer: = (3r + 2)(3r - 2)(4r^2 + 5) Hide answer | Show step-by-step
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4.8 Factoring Expressions with Fractional or Negative Exponents

Example 1: factoring an expression with fractional or negative exponents.

  • Factor [latex]3x{(x\;+\;2)}^\frac{-1}3\:+\;4{(x\:+\:2)}^\frac23[/latex].
  • Factor [latex]2\left(5a-1\right)^\frac{3}{4}+7a\left(5a-1\right)^\frac{-1}{4}[/latex]

Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

Math 3080 Preparation Copyright © 2022 by Erin Kox is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Exponents Rules Worksheets 99Worksheets

Factoring Exponents Worksheets

Factoring Exponents Worksheets – Factor worksheets are a must-have tool to teach and learn about factors, prime numbers and multiplication. They help students achieve a thorough understanding of these fundamental mathematical concepts while providing teachers with a useful assessment tool. In this comprehensive guide, we’ll review different types of factor worksheets. They will provide step by step instructions on how to make your own, and offer guidelines for teaching factors effectively.

What are Factor Worksheets?

Factor worksheets are printable pages designed to help students practice discovering those factors that make up numbers as well as identifying prime numbers and understanding the relationship between multiplication and division. They are often a combination of activities that require learners to list their factors, determine the greatest common factor (GCF) and to perform prime factorization.

10 Worksheets For 8Th Grade Coo Worksheets

Types of Factor Worksheets:

A. Factor Tree Worksheets

Factor tree worksheets guide students through learning how they break down numbers into their primary factors using an arrangement that resembles a tree. This visual approach aids students recognize the most important elements of numbers. It also helps in the process of discovering the largest common factor (or small number of common factors).

B. Greatest Common Factor Worksheets

Greatest common factor worksheets concentrate in helping students recognize what is the most important factor shared by 2 or more figures. They usually include tasks that require students to record factors, compare them, and calculate the GCF.

C. Prime Factorization Worksheets

Prime factorization worksheets are designed to teach students to break down number into their prime factors employing various techniques, like factor tree, division or the upside-down cake method. These worksheets help students understand the elements that make up numbers which will help them develop their multiplication and division skills.

Factoring By Grouping Worksheet Algebra 2 Answers With Work Kathlyn

How to Create Factor Worksheets:

A. Choose the Right Template

Choose a template to fit the kind of worksheet you’d like you to create for example, factor trees, the greatest common factor, or prime factors. You can find free templates online or create custom ones with word processing software.

B. Customize the Content

Customize the content of the worksheet to your students’ interests and levels. Include a variety of simple medium, challenging, and easy problems that challenge and engage students. Be sure that your instructions are precise and concise, to ensure that students are aware of what is expected of them.

C. Include Answer Keys

Create an answer sheet for each workbook to allow students to check their work and aid teachers with grading. This is especially helpful for more complicated tasks that involve multiple steps.

Exponents Rules Worksheets 99Worksheets

Tips for teaching Factors with Factor Worksheets

  • Begin with concrete examples: Begin by teaching the concepts through real-world scenarios, such as the grouping of objects or using arrays, in order to help students to build a solid foundation to understand the factors.
  • Utilize manipulatives. students to use tangible or digital manipulatives, whether they are used to learn about factors and prime numbers. This is because it can assist students in understanding concepts more effectively.
  • Use factor-related vocabulary to teach: Make sure students are aware of the terms related to factors, such as prime, composite, GCF, and LCM which will assist students better communicate their understanding of concepts.
  • Introduce different learning styles: Employ a variety of methods for teaching that include instructions in the form of direct instruction, group games, and individual practice, so as to accommodate different ways of learning and keep your students engaged.
  • Monitor learning: Regularly review students’ progress through quizzes, tests, and other class work to spot areas where they may require additional help or assistance.
  • Encourage self-assessment. Create the development mindset when students are encouraged to evaluate their work and discover the areas where they could improve. This will allow them to develop critical thinking abilities and make learning their own.

Conclusion:

Factor worksheets are a powerful tool to teach and learn about factors, prime numbers and multiplication. When you are aware of the various types of worksheets that are available for factor learning as well as creating custom content as well as implementing effective teaching strategies that help students gain a firm foundation in the fundamental math concepts. By putting in the effort and persevering students will be able to acquire the knowledge and skills they need to excel in math.

Free Factor Worksheet Templates:

To assist you obtain commenced getting started, we’ve collected a selection of factor worksheets for free for you to download and utilize as a classroom resource. The worksheets cover a range of subjects, including factor trees, greatest common factors, as well as prime factorization. Click on the links lower to download and print the worksheets:

  • Factor Tree Worksheets
  • Greatest Common Factor Worksheets
  • Prime Factorization Worksheets

We hope that this guide has provided valuable insight into the world in factor workbooks and ways that they can be utilized to increase your students’ understanding of prime numbers, factors, and multiplication. We wish you a happy teaching!

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Grade 5 Factoring Worksheets

Factoring, gcf and lcm worksheets.

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Prime Factorization Worksheets

Prime Factorization Worksheet image

Welcome to our Prime Factorization Worksheets page.

Here you will find our selection of worksheets involving finding prime factorizations for a range of numbers, and for solving prime factor problems.

We have a range of sheets at different levels of difficulty.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Using these factor tree worksheets will help your child to:

  • practice prime factorization of a range of numbers;
  • solve problems involving prime factorization;
  • use and understand exponential notation.

These sheets are aimed at students from 6th grade and upwards.

We also have a link to our Prime Factorization Calculator which will quickly and easily show you all the prime factors of any number.

There is also a quick quiz at the bottom of the page where you can test your skill at prime factorization and get immediate feedback!

What is prime factorization?

To find out more about prime factorization, including how it works and to look at some worked examples, take a look at our prime factorization support page.

  • What is Prime Factorization support page

Examples of Prime Factorization

Example 1) Find the prime factorization of 40.

40 is even, so we know that 2 is a factor (as well as being prime)

40 = 2 x 20.

Now we move on to the number 20.

Again 20 is even, so 2 is a factor.

20 = 2 x 10

So 40 = 2 x 20 = 2 x 2 x 10

Now we factorize 10 into 2 x 5 (both prime numbers) and we cannot go any further, as all the numbers are prime.

So 40 = 2 x 20 = 2 x 2 x 10 = 2 x 2 x 2 x 5

Our final answer is 40 = 2 x 2 x 2 x 5 = 2 3 x 5

Example 2) Find the prime factorization of 45.

45 is odd, so 2 is not a factor.

As the number ends in a 5 we know that 5 is a factor.

45 = 5 x 9.

Now we have to factorize 9.

9 = 3 x 3 (which is a prime number).

This gives us:

45 = 5 x 9 = 5 x 3 x 3

Our final answer is 45 = 3 x 3 x 5 = 3 2 x 5

Example 3) Find the prime factorization of 126.

126 is even, so 2 is a factor.

126 = 2 x 63

Now we need to factorize 63.

63 = 7 x 9.

7 is prime, but 9 is not

126 = 2 x 63 = 2 x 7 x 9 = 2 x 7 x 3 x 3

Our final answer is 126 = 2 x 3 x 3 x 7 = 2 x 3 2 x 7

All the worksheets have answers provided. The answers are given with the factors in ascending order.

The sheets get gradually harder in each section.

We have split the sheets into 3 different sections:

  • Our first section involves straightforward prime factorization of different numbers.
  • Our next group of sheets involves solving prime factorization riddles.
  • Our final group of sheets involves finding all possibilities in our prime factorization challenges.

Prime Factorization Worksheets - 1st Section

  • The two sheets just has straightforward prime factorization of a range of numbers.
  • The third sheet is harder, and involves using exponential notation and looking at prime factorization from different starting points
  • Prime Factorization Sheet 1
  • PDF version
  • Prime Factorization Sheet 2
  • Prime Factorization Sheet 3

Prime Factorization Worksheets - Riddles - 2nd Section

These sheets involve solving prime factorization riddles.

There are 2 riddles on each sheet and 8 possible answers.

By correctly working out the clues in each riddle, all the incorrect answers are eliminated leaving just one correct answer.

  • Prime Factorization Riddles Sheet 1
  • Prime Factorization Riddles Sheet 2

Prime Factorization Worksheets - Challenges - 3rd Section

Our prime factorization challenges are a great way to stretch more able students.

The challenges involve finding all possible numbers that have certain features involving prime factors.

  • Prime Factorization Challenges 1
  • Prime Factorization Challenges 2

Prime Factorization Walkthrough Video

This short video walkthrough shows several problems from our Prime Factorization Worksheet 3 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on this page, check out the video!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

  • Factors and Multiples Worksheets

We have worksheets to help you understand more about factors and multiples.

They are aimed at a 4th/5th grade level.

  • Prime Factorization Calculator

Our prime factorization calculator will help you factor any number into a product of its prime factors.

It will show the answer in exponential form as well as standard form.

You can use this calculator to check that your prime factors in your factor trees are correct.

Prime Factorization Calculator Picture

  • Factor Tree Worksheets

Using factor trees is a great visual way to understand prime factorization.

The worksheets on the link below are at an easier level to the sheets on this page and help children learn all about factor trees and prime factors.

  • Greatest Common Factor Worksheets

As well as our calculator, we also have a range of greatest common factor worksheets.

These sheets have been designed for 6th and 7th grade students.

They will help students learn and practice finding the greatest common factor of numbers up to 100.

  • Least Common Multiple Worksheets

We have a range of worksheets on how to find the least common multiple of two or three numbers.

The sheets vary in difficulty, and are suitable for 6th grade and up.

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

  • Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

  • What is a Prime Number

Divisibility Rules 1-10 Chart

We have a range of charts which can help you determine whether a number between 1 and 10 is a factor of a number.

  • Divisibility Rules 1-10 Charts

6th Grade Mental Math Quizzes

Here you will find a range of printable mental math 6th grade quizzes for your child to enjoy.

Each worksheet tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math quiz!

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Prime Factorization Online Quiz

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

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For more information on the information we collect, please take a look at our Privacy Policy

We would be grateful for any feedback on our quizzes, please let us know using our Contact Us link, or use the Facebook Comments form at the bottom of the page.

This quick quiz tests your knowledge of prime factorization with numbers up to 120.

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Key to Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language, and examples are easy to follow. Word problems relate algebra to familiar situations, helping students to understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system.

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IMAGES

  1. Factoring Exponents Worksheets

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  2. Exponents Review Worksheet

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  3. Exponents Worksheets Grade 7

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  4. the worksheet for multiplying exponents is shown in black and white

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  5. Exponents in Factor Form (A)

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  6. 13 Powers And Exponents Worksheet / worksheeto.com

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VIDEO

  1. Exponents & Factors in Math Explained

  2. Factoring Polynomials

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  4. Algebra 2 Factoring Worksheet 1

  5. CLASS 8..EXPONENTS AND POWERS PRACTICE WORKSHEET 🤗🤗#education #shortsvideo #ytshorts

  6. Factoring Practice

COMMENTS

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    Welcome to The Exponents in Factor Form (A) Math Worksheet from the Number Sense Worksheets Page at Math-Drills.com. This math worksheet was created or last revised on 2008-08-18 and has been viewed 83 times this week and 129 times this month.

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    Free Algebra 2 worksheets created with Infinite Algebra 2. Printable in convenient PDF format. Kuta Software. ... Factoring a sum/difference of cubes; Factoring by grouping ... Radical Functions and Rational Exponents. Simplifying radicals; Operations with radical expressions ...

  4. PDF 7.1B Rational Exponents II

    . Reducing fractions, we get Ans Example 2: Factor the GCF out of . Solution Write each fraction in terms of the LCD of the exponents, to be able to identify the smallest power. : Thus . Thus we have . Since the smallest power is , we have a GCF of . Thus we have . Reducing fractions, we get Ans

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    KSU Important Properties: 2 Di®erence of squares: a2 ¡ b2 = (a ¡ b)(a + b) 2 Di®erence of cubes: a3 ¡ b3 = (a ¡ b)(a2 + ab + b2) 2 Sum of cubes: a3 + b3 = (a + b)(a2 ¡ ab + b2) 2 Remember that you can always check your factoring by multiplication. 2 Always look for the greatest common factor (GCF) ̄rst.

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    Quiz Unit test About this unit Expand your algebra superpowers by introducing exponents! Let's build our toolkit that allows us to manipulate exponents algebraically. Exponent properties intro Learn Exponent properties with products Exponent properties with parentheses Exponent properties with quotients Exponent properties review Practice

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    Factoring Expressions with Exponents Definition: To factor a polynomial is to write the addition of two or more terms as the product of two or more terms. Consider the addition of the two numbers 24 + 30. Notice that they are both multiples of 6. We could write The factors are '6' and ' (4+5)'.

  8. 1.5: Factoring Polynomials

    These polynomials are said to be prime. Howto: Given a trinomial in the form x2 + bx + c x 2 + b x + c, factor it. Find p p and q q, a pair of factors of c c with a sum of b b. Write the factored expression (x + p)(x + q) ( x + p) ( x + q). Example 1.5.2 1.5. 2: Factoring a Trinomial with Leading Coefficient 1.

  9. PDF Worksheet 2 6 Factorizing Algebraic Expressions

    Worksheet 2:6 Factorizing Algebraic Expressions Section 1 Finding Factors Factorizing algebraic expressions is a way of turning a sum of terms into a product of smaller ones. The product is a multiplication of the factors. Sometimes it helps to look at a simpler case before venturing into the abstract. The number 48 may be written as a product in a

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    Variables With Negative Exponents Worksheet; Slope, Gradient, and Slope Intercept. Determining Slope From a Graph Worksheet; Solve by Factoring Lessons. ... Factoring Completely Worksheet; Factoring Completely. Factor each expression completely. Checking Your Answers.

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    Instructions: Choose an answer and hit 'next'. You will receive your score and answers at the end. question 1 of 3 Find the greatest common factor of the expression below: Next Worksheet Print...

  14. 4.8 Factoring Expressions with Fractional or Negative Exponents

    Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, 2x1 4 + 5x3 4 ...

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    Exponents Worksheets Exponents and Powers of Ten Our exponents worksheets cover reading and writing simple exponents, powers of ten, whole number, fractional and decimal bases, negative exponents and equations with exponents Choose your grade / topic: Grade 5 exponents worksheets Introduction to exponents Reading and writing simple exponents

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    1 Factoring and Rational Expressions Monday: { Topic: Review of Factoring (5.6) { Group work and worksheets p. 2-4 { Study skills: Assessment worksheet { Homework: 5.6 Tuesday: { Topic: Rational Expressions and Rational Functions (6.1) { Quiz: 5.6 { Worksheets p. 5-6 { Homework: 6.1 Wednesday: { Topic: Adding Rational Expressions (6.2(a))

  18. Factoring Worksheets

    Grade 5 factoring worksheets. Factoring numbers to prime factors (up to 100, 500) List all the factors of numbers up to 100. Prime factor trees. More divisibility rules. Greatest common factor (GCF) of 2 numbers (1-50, 1-100) Lowest common multiple (LCM) of 2 numbers 2-30. GCF and LCM word problems.

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    Factoring Exponents Worksheets - Factor worksheets are a must-have tool to teach and learn about factors, prime numbers and multiplication. They help students achieve a thorough understanding of these fundamental mathematical concepts while providing teachers with a useful assessment tool.

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  22. Prime Factorization Worksheet page

    Example 1) Find the prime factorization of 40. 40 is even, so we know that 2 is a factor (as well as being prime) 40 = 2 x 20. Now we move on to the number 20. Again 20 is even, so 2 is a factor. 20 = 2 x 10 So 40 = 2 x 20 = 2 x 2 x 10 Now we factorize 10 into 2 x 5 (both prime numbers) and we cannot go any further, as all the numbers are prime.

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    With this worksheet generator, you can make customizable worksheets for the distributive property and factoring. These worksheets are especially meant for pre-algebra and algebra 1 courses (grades 6-9). The generator includes only very simple problems with linear expressions. We hope to add more problem types later. Basic instructions for the ...