algebra 1 assignment factor each completely

Factoring Completely

Factor each expression completely.

Checking Your Answers

Enter an answer in each box, then click the “Show Answers” button at the bottom of the page to see the answers.

If you need assistance with a particular problem, click the “step-by-step” link for an in depth solution.

• Equation: x^3 - x Show answer | Show step-by-step Answer: = x(x - 1)(x + 1) Hide answer | Show step-by-step
• Equation: x^3 + 5x^2 + 6x Show answer | Show step-by-step Answer: = x(x + 2)(x + 3) Hide answer | Show step-by-step
• Equation: 4y^2 - 8y - 60 Show answer | Show step-by-step Answer: = 4(y + 3)(y - 5) Hide answer | Show step-by-step
• Equation: 3q^2 - 12q + 12 Show answer | Show step-by-step Answer: = 3(q - 2)(q - 2) Hide answer | Show step-by-step
• Equation: 3m^3 + 33m^2 + 90m Show answer | Show step-by-step Answer: = 3m(m + 5)(m + 6) Hide answer | Show step-by-step
• Equation: 4x^2 + 8xy + 4y^2 Show answer | Show step-by-step Answer: = 4(x + y)(x + y) Hide answer | Show step-by-step
• Equation: 4kx^2 - 4ky^2 Show answer | Show step-by-step Answer: = 4k(x + y)(x - y) Hide answer | Show step-by-step
• Equation: m^4 - 1 Show answer | Show step-by-step Answer: = (m + 1)(m - 1)(m^2 + 1) Hide answer | Show step-by-step
• Equation: p^4 - 2p^2 + 1 Show answer | Show step-by-step Answer: = (p - 1)(p + 1)(p + 1)(p - 1) Hide answer | Show step-by-step
• Equation: abc^2 + 6abc + 5ab Show answer | Show step-by-step Answer: = ab(c + 1)(c + 5) Hide answer | Show step-by-step
• Equation: 24x^2 + 68x + 48 Show answer | Show step-by-step Answer: = 4(2x + 3)(3x + 4) Hide answer | Show step-by-step
• Equation: 4x^2y + 12xy^2 + 9y^3 Show answer | Show step-by-step Answer: = y(2x + 3y)(2x + 3y) Hide answer | Show step-by-step
• Equation: 3mx^2 - 5mx - 28m Show answer | Show step-by-step Answer: = m(3x + 7)(x - 4) Hide answer | Show step-by-step
• Equation: 21k - 98k^2 - 343k^3 Show answer | Show step-by-step Answer: = 7k(1 - 7k)(3 + 7k) Hide answer | Show step-by-step
• Equation: 4p^4 - 5p^2 + 1 Show answer | Show step-by-step Answer: = (2p + 1)(2p - 1)(p + 1)(p - 1) Hide answer | Show step-by-step
• 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
• Equation: 108s^6 + 42s^4 - 6s^2 Show answer | Show step-by-step Answer: = 6s^2(3s + 1)(3s - 1)(2s^2 + 1) Hide answer | Show step-by-step
• Equation: 144uv^4 - 388uv^2 + 144u Show answer | Show step-by-step Answer: = 4u(2v + 3)(2v - 3)(3v + 2)(3v - 2) Hide answer | Show step-by-step
• Equation: 6m^3n - 9m^2n^2 - 15mn^3 Show answer | Show step-by-step Answer: = 3mn(m + n)(2m - 5n) Hide answer | Show step-by-step
• Equation: 2p^3x + p^2x - 28px Show answer | Show step-by-step Answer: = px(p + 4)(2p - 7) Hide answer | Show step-by-step

Related Pages

Factoring Completely Lesson Brush up on your knowledge of the techniques needed to solve problems on this page.

Expression Factoring Calculator Simplifies and factors expressions automatically.

Looking for someone to help you with algebra? At Wyzant, connect with algebra tutors and math tutors nearby. Prefer to meet online? Find online algebra tutors or online math tutors in a couple of clicks.

• The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds
• Nine subtracted from w is greater than or equal to -22
• In a game, you have a 1/22 probability of winning $135 and a 21/22 probability of losing $10. What is your expected winning?
• Help me answer this

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

7.5: General Strategy for Factoring Polynomials

• Last updated
• Save as PDF
• Page ID 15170

Learning Objectives

By the end of this section, you will be able to:

• Recognize and use the appropriate method to factor a polynomial completely

Before you get started, take this readiness quiz.

• Factor \(y^{2}-2 y-24\). If you missed this problem, review Example 7.2.19 .
• Factor \(3 t^{2}+17 t+10\). If you missed this problem, review Example 7.3.28 .
• Factor \(36 p^{2}-60 p+25\). If you missed this problem, review Example 7.4.1 .
• Factor \(5 x^{2}-80\). If you missed this problem, review Example 7.4.31 .

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. Figure \(\PageIndex{1}\) outlines a strategy you should use when factoring polynomials.

FACTOR POLYNOMIALS.

• Factor it out.
• Of squares? Sums of squares do not factor.
• Of cubes? Use the sum of cubes pattern.
• Of squares? Factor as the product of conjugates.
• Of cubes? Use the difference of cubes pattern.
• If aa and cc are squares, check if it fits the trinomial square pattern.
• Use the trial and error or “ac” method.
• If it has more than three terms: Use the grouping method.
• Is it factored completely?
• Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example \(\PageIndex{1}\)

Factor completely: \(4 x^{5}+12 x^{4}\)

\(\begin{array}{lll} \text { Is there a GCF? } & \text { Yes, } 4 x^{4}& 4 x^{5}+12 x^{4} \\ \text { Factor out the GCF. } & &4 x^{4}(x+3) \\ \text { In the parentheses, is it a binomial, a } & & \\ \text { trinomial, or are there more than three terms? } & \text { Binomial. } & \\ \quad \text { Is it a sum? } & & \text { Yes. } \\ \quad \text { Of squares? Of cubes? } & & \text { No. }\\ \text { Check. } \\ \\ \quad \text { Is the expression factored completely? } & & \text{ Yes.} \\ \quad \text { Multiply. } \\ \begin{array}{l}{4 x^{4}(x+3)} \\ {4 x^{4} \cdot x+4 x^{4} \cdot 3} \\ {4 x^{5}+12 x^{4}}\checkmark \end{array}\end{array}\)

Try It \(\PageIndex{2}\)

Factor completely: \(3 a^{4}+18 a^{3}\)

3\(a^{3}(a+6)\)

Try It \(\PageIndex{3}\)

Factor completely: \(45 b^{6}+27 b^{5}\)

9\(b^{5}(5 b+3)\)

Example \(\PageIndex{4}\)

Factor completely: \(12 x^{2}-11 x+2\)

Try It \(\PageIndex{5}\)

Factor completely: \(10 a^{2}-17 a+6\)

\((5 a-6)(2 a-1)\)

Try It \(\PageIndex{6}\)

Factor completely: \(8 x^{2}-18 x+9\)

\((2 x-3)(4 x-3)\)

Example \(\PageIndex{7}\)

Factor completely: \(g^{3}+25 g\)

\(\begin{array}{lll} \text { Is there a GCF? } & \text{Yes, g.} &g^{3}+25 g \\\text { Factor out the GCF. } & &g\left(g^{2}+25\right) \\ \text { In the parentheses, is it a binomial, trinomial, } & & \\ \text { or are there more than three terms? } &\text { Binomial. } & \\ \quad \text { Is it a sum? Of squares? } & \text { Yes. } & \text { Sums of squares are prime. } \\\text { Check. } \\ \\ \quad \text { Is the expression factored completely? } &\text { Yes. } \\ \quad \text { Multiply. } \\ \qquad \begin{array}{l}{g\left(g^{2}+25\right)} \\ {g^{3}+25 g }\checkmark \end{array} \end{array}\)

Try It \(\PageIndex{8}\)

Factor completely: \(x^{3}+36 x\)

\(x\left(x^{2}+36\right)\)

Try It \(\PageIndex{9}\)

Factor completely: \(27 y^{2}+48\)

3\(\left(9 y^{2}+16\right)\)

Example \(\PageIndex{10}\)

Factor completely: \(12 y^{2}-75\)

\(\begin{array}{lll} \text { Is there a GCF? } & \text{Yes, 3.} &12 y^{2}-75 \\\text { Factor out the GCF. } & &3\left(4 y^{2}-25\right) \\ \text { In the parentheses, is it a binomial, trinomial, } & & \\ \text { or are there more than three terms? } &\text { Binomial. } & \\ \text { Is it a sum?} & \text { No. } & \\ \text { Is it a difference? Of squares or cubes? } &\text { Yes, squares. } & 3\left((2 y)^{2}-(5)^{2}\right) \\ \text { Write as a product of conjugates. } & &3(2 y-5)(2 y+5)\\\text { Check. } \\ \\ \text { Is the expression factored completely? } & \text{ Yes.}& \\ \text { Neither binomial is a difference of } \\ \text { squares. } \\ \text{ Multiply.} \\ \quad \begin{array}{l}{3(2 y-5)(2 y+5)} \\ {3\left(4 y^{2}-25\right)} \\ {12 y^{2}-75}\checkmark \end{array} \end{array}\)

Try It \(\PageIndex{11}\)

Factor completely: \(16 x^{3}-36 x\)

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

Try It \(\PageIndex{12}\)

Factor completely: \(27 y^{2}-48\)

3\((3 y-4)(3 y+4)\)

Example \(\PageIndex{13}\)

Factor completely: \(4 a^{2}-12 a b+9 b^{2}\)

Try It \(\PageIndex{14}\)

Factor completely: \(4 x^{2}+20 x y+25 y^{2}\)

\((2 x+5 y)^{2}\)

Try It \(\PageIndex{15}\)

Factor completely: \(9 m^{2}+42 m n+49 n^{2}\)

\((3 m+7 n)^{2}\)

Example \(\PageIndex{16}\)

Factor completely: \(6 y^{2}-18 y-60\)

\(\begin{array}{lll} \text { Is there a GCF? } & \text{Yes, 6.} &6 y^{2}-18 y-60 \\\text { Factor out the GCF. } & \text { Trinomial with leading coefficient } 1&6\left(y^{2}-3 y-10\right) \\ \text { In the parentheses, is it a binomial, trinomial, } & & \\ \text { or are there more terms? } & & \\ \text { "Undo' FOIL. } & 6(y\qquad )(y\qquad ) &6(y+2)(y-5) \\ \text { Check your answer. } \\ \text { Is the expression factored completely? } & & \text{ Yes.} \\ \text { Neither binomial is a difference of squares. } \\ \text { Multiply. } \\ \\\qquad \begin{array}{l}{6(y+2)(y-5)} \\ {6\left(y^{2}-5 y+2 y-10\right)} \\ {6\left(y^{2}-3 y-10\right)} \\ {6 y^{2}-18 y-60} \checkmark \end{array} \end{array}\)

Try It \(\PageIndex{17}\)

Factor completely: \(8 y^{2}+16 y-24\)

8\((y-1)(y+3)\)

Try It \(\PageIndex{18}\)

Factor completely: \(5 u^{2}-15 u-270\)

5\((u-9)(u+6)\)

Example \(\PageIndex{19}\)

Factor completely: \(24 x^{3}+81\)

Try It \(\PageIndex{20}\)

Factor completely: \(250 m^{3}+432\)

2\((5 m+6)\left(25 m^{2}-30 m+36\right)\)

Try It \(\PageIndex{21}\)

Factor completely: \(81 q^{3}+192\)

\(3(3q+4)\left(9q^{2}-12 q+16\right)\)

Example \(\PageIndex{22}\)

Factor completely: \(2 x^{4}-32\)

\(\begin{array}{llc} \text { Is there a GCF? } & \text{Yes, 2.} &2 x^{4}-32 \\\text { Factor out the GCF. } & &2\left(x^{4}-16\right) \\ \text { In the parentheses, is it a binomial, trinomial, } & & \\ \text { or are there more than three terms? } & \text { Binomial. }& \\ \text { Is it a sum or difference? } &\text { Yes. }& \\\text { Of squares or cubes? } & \text { Difference of squares. } & 2\left(\left(x^{2}\right)^{2}-(4)^{2}\right) \\ \text { Write it as a product of conjugates. } & & 2\left(x^{2}-4\right)\left(x^{2}+4\right) \\ \text { The first binomial is again a difference of squares. } & & 2\left((x)^{2}-(2)^{2}\right)\left(x^{2}+4\right) \\ \text { Write it as a product of conjugates. } & & 2(x-2)(x+2)\left(x^{2}+4\right) \\ \text { Is the expression factored completely? } &\text { Yes. } & \\ \\ \text { None of these binomials is a difference of squares. } \\ \text { Check your answer. } \\ \text{ Multiply. }\\ \\ \qquad \qquad \begin{array}{l}{2(x-2)(x+2)\left(x^{2}+4\right)} \\ {2(x-2)(x+2)\left(x^{2}+4\right)} \\ {2(x-10)} \\ {2 x^{4}-32} \checkmark \end{array} \end{array}\)

Try It \(\PageIndex{23}\)

Factor completely: \(4 a^{4}-64\)

4\(\left(a^{2}+4\right)(a-2)(a+2)\)

Try It \(\PageIndex{24}\)

Factor completely: \(7 y^{4}-7\)

7\(\left(y^{2}+1\right)(y-1)(y+1)\)

Example \(\PageIndex{25}\)

Factor completely: \(3 x^{2}+6 b x-3 a x-6 a b\)

\(\begin{array}{llc} \text { Is there a GCF? } & \text{Yes, 3.} &3 x^{2}+6 b x-3 a x-6 a b\\\text { Factor out the GCF. } & &3\left(x^{2}+2 b x-a x-2 a b\right)\\ \text { In the parentheses, is it a binomial, trinomial, } &\text { More than } 3 & \\ \text { or are there more terms? } &\text { terms. } & \\ \text { Use grouping. } & & \begin{array}{c}{3[x(x+2 b)-a(x+2 b)]} \\ {3(x+2 b)(x-a)}\end{array} \\ \text { Check your answer. } \\ \\ \text { Is the expression factored completely? Yes. } \\ \text { Multiply. } \\\qquad \qquad \begin{array}{l}{3(x+2 b)(x-a)} \\ {3\left(x^{2}-a x+2 b x-2 a b\right)} \\ {3 x^{2}-3 a x+6 b x-6 a b} \checkmark \end{array}\end{array}\)

Try It \(\PageIndex{26}\)

Factor completely: \(6 x^{2}-12 x c+6 b x-12 b c\)

6\((x+b)(x-2 c)\)

Try It \(\PageIndex{27}\)

Factor completely: \(16 x^{2}+24 x y-4 x-6 y\)

2\((4 x-1)(x+3 y)\)

Example \(\PageIndex{28}\)

Factor completely: \(10 x^{2}-34 x-24\)

\(\begin{array}{llc} \text { Is there a GCF? } & \text{Yes, 2.} &10 x^{2}-34 x-24\\\text { Factor out the GCF. } & &2\left(5 x^{2}-17 x-12\right)\\ \text { In the parentheses, is it a binomial, trinomial, } &\text { Trinomial with } & \\ \text { or are there more than three terms? } &\space a \neq 1 & \\ \text { Use trial and error or the "ac" method. } & & 2\left(5 x^{2}-17 x-12\right) \\ & & 2(5 x+3)(x-4) \\ \text { Check your answer. Is the expression factored } \\\text { completely? Yes. }\\ \\ \text { Multiply. } \\ \qquad \begin{array}{l}{2(5 x+3)(x-4)} \\ {2\left(5 x^{2}-20 x+3 x-12\right)} \\ {2\left(5 x^{2}-17 x-12\right)} \\ {10 x^{2}-34 x-24}\checkmark \end{array}\end{array}\)

Try It \(\PageIndex{29}\)

Factor completely: \(4 p^{2}-16 p+12\)

4\((p-1)(p-3)\)

Try It \(\PageIndex{30}\)

Factor completely: \(6 q^{2}-9 q-6\)

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

Key Concepts

• General Strategy for Factoring Polynomials See Figure \(\PageIndex{1}\).
• Is there a greatest common factor? Factor it out.
• If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
• Use the trial and error or ‘ac’ method.
• Check. Is it factored completely? Do the factors multiply back to the original polynomial?

Algebra 1 Factor Each Completely Worksheet

Algebra 1 Factor Each Completely Worksheet – Factor worksheets provide a vital tool for teaching and learning about factors, prime numbers and multiplication. These printable worksheets assist students to develop a strong understanding of the basic mathematical concepts and provide teachers with a effective assessment tool. In this extensive guide, we’ll look at different kinds of factor worksheets. They will provide step by step instructions on how to make your own, and provide tips for teaching factors effectively.

What are Factor Worksheets?

Factor worksheets can be printed as sheets designed to help students practice understanding factorization of number and identify prime numbers and to understand the connection between division and multiplication. They typically include a variety of tasks that require students to list factors, find the greatest common factor (GCF) as well as perform prime factorization.

Types of Factor Worksheets:

A. Factor Tree Worksheets

Factor tree worksheets guide students through their process of breaking numbers down to their prime factors, using this tree-like design. This visual approach aids students discover the prime factors of the numbers and simplify the process of finding the most common factor, which is also known as a small-common multiple.

B. Greatest Common Factor Worksheets

Greatest Common Factor worksheets focus on helping students to identify the factor with the highest share of at least two numbers. These worksheets typically contain questions which require students to write down the factors they are able to list, compare them as well as determine the GCF.

C. Prime Factorization Worksheets

Prime factorization worksheets help students how to break down the number into prime factors by using a variety of methods, including factor trees, division, or the upside-down cake method. These worksheets will help students comprehend the foundational elements of numbers in order to enhance their multiplication and division skills.

How to Create Factor Worksheets:

A. Choose the Right Template

Choose a template that matches the kind of worksheet you want to create like factor trees, greatest common factor, or prime factorization. Find free templates online or design ones using word processing programs.

B. Customize the Content

You can tailor the content of the worksheet to meet your students’ interests and levels. Include a variety of simple moderate, difficult, and questions to engage and challenge students. Be sure that your instructions are clear and concise, so students know what is required of them.

C. Include Answer Keys

Create an answer key for every worksheet that helps students assess their work and assist teachers in grading. This can be especially useful in more difficult problems that involve multiple steps.

Tips for teaching Factors with Factor Worksheets

• Begin with concrete examples: Start with real-world situations, like grouping objects , or using arrays, in order to help students gain a solid basis for understanding aspects.
• Use manipulativesand encourage students to utilize both digital and physical manipulatives in their exploration of prime number, which can help students visualize the concepts more effectively.
• Instruct factor vocabulary You must ensure that your students understand the terms related to factors, like prime composite, GCF also known as LCM, as this will help them more effectively communicate their understanding of concepts.
• Incorporate different learning methods: Use a variety of teaching strategies which include groups, direct instruction and individual exercises, in order to meet different methods of learning, and keep the students at their best.
• Monitor the students’ progress regularly: Observe their’ progress during quizzes tests, and other class work to find areas where they might require extra help or additional practice.
• Encourage self-assessment. This can foster an attitude of improvement by encouraging students evaluate the quality of their work and to identify the areas where they could improve. This will allow them to develop critical thinking abilities and accept responsibility for their learning.

Conclusion:

Factor worksheets provide a great method of teaching and learning about prime numbers, factors, and multiplication. When you are aware of the various types of factor worksheets available making custom content and implementing effective methods for teaching teachers can help students build a solid foundation for these fundamental mathematical concepts. By putting in the effort and persevering students will be able to acquire those skills and attitudes that will help them succeed in math.

Free Factor Worksheet Templates:

To help you start beginning, we’ve designed a series of free factor worksheet templates that you can download and use within your classes. These worksheets cover a variety of topics, such as factor treesas well as the most popular factors, as well as prime factorization. Click on the links low to access and print the worksheets:

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

We hope that this complete guide has helped you gain insight into the concept of factor worksheets and how they can be used to increase your students’ knowledge of factors, prime numbers, and multiplication. Happy teaching!

Gallery of Algebra 1 Factor Each Completely Worksheet

Leave a Comment Cancel reply

Save my name, email, and website in this browser for the next time I comment.