Easy Steps: How to Factor Polynomials and Solve Problems

factoring and dividing with polynomials

Subjects

Algebra 2

Dividing Polynomials & Quadratics

Dividing Polynomials by Linear Factors

448

•

Jun 12, 2023

•

1 page

Easy Steps: How to Factor Polynomials and Solve Problems

Name: Knowunity
Brand: Knowunity

Mira

@miraa._noui

This document provides an overview of factoring polynomials and polynomial... Show more

(
finol study guide
1. factoring:
→ always factor out the gef first, if there is one
a. difference of squares: a²_b² → (a+b) (a-b)
La
FACTOR

Factoring Techniques

This section covers various methods for factoring polynomials with different numbers of terms, including:

Factoring out the Greatest Common Factor $GCF$
Difference of Squares
Sum or Difference of Cubes
Factoring Quadratics
Factoring by Grouping
Other Factoring Techniques

Highlight: Always factor out the Greatest Common Factor $GCF$ first, if there is one.

Example: For difference of squares: a² - b² → $a+b$ $a-b$

Vocabulary: Difference of Squares - A special case of factoring where a polynomial is in the form a² - b².

Example: For sum of cubes: a³ + b³ → $a+b$ $a² - ab + b²$

Example: For difference of cubes: a³ - b³ → $a-b$ $a² + ab + b²$

Highlight: When factoring quadratics $ax² + bx + c$ , find two factors of ac that add up to b.

Example: Factoring x² + 8x + 12:

Factors of 12 that add up to 8 are 6 and 2
Rewrite as x² + 6x + 2x + 12
Factor by grouping: x $x+6$ + 2 $x+6$
Final result: $x+2$ $x+6$

Vocabulary: Factor by Grouping - A method used to factor polynomials with four or more terms by separating them into groups.

Polynomial Long Division

This section explains the process of polynomial long division, which is similar to regular long division but with polynomials.

Highlight: Don't forget to use placeholders for any missing degrees in the polynomial.

Example: Dividing $2x² + x - 3$ by $x + 1$ :

Divide first terms: 2x² ÷ x = 2x
Multiply: 2x $x + 1$ = 2x² + 2x
Subtract: 2x² + x - 3 - $2x² + 2x$ = -x - 3
Repeat the process with -x - 3
Final result: 2x - 1 with a remainder of -2

Synthetic Division

The guide concludes with an explanation of synthetic division, a shortcut method for dividing polynomials.

Vocabulary: Synthetic Division - A simplified method of polynomial long division, particularly useful when dividing by a linear factor.

Example: Synthetic division of 2x⁴ - 7x³ - 7x + 1 by $x - 4$ :

Write coefficients: 2 -7 0 -7 1
Use 4 as the divisor $root of x - 4$
Bring down first coefficient: 2
Multiply 2 by 4 and add to -7: 2 $4$ + $-7$ = 1
Continue this process for all terms
Final result: 2x³ + x² + 4x + 9 with a remainder of 37

Highlight: Use zero as a placeholder for missing terms in the polynomial during synthetic division.

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Subjects

Algebra 2

Dividing Polynomials & Quadratics

Dividing Polynomials by Linear Factors

Algebra 2

•

448

•

Jun 12, 2023

•

1 page

Easy Steps: How to Factor Polynomials and Solve Problems

Mira

@miraa._noui

This document provides an overview of factoring polynomials and polynomial division techniques. It covers various methods for factoring, including difference of squares, sum or difference of cubes, quadratic factoring, and grouping. The guide also explains polynomial long division and synthetic... Show more

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Factoring Techniques

This section covers various methods for factoring polynomials with different numbers of terms, including:

Factoring out the Greatest Common Factor $GCF$
Difference of Squares
Sum or Difference of Cubes
Factoring Quadratics
Factoring by Grouping
Other Factoring Techniques

Highlight: Always factor out the Greatest Common Factor $GCF$ first, if there is one.

Example: For difference of squares: a² - b² → $a+b$ $a-b$

Vocabulary: Difference of Squares - A special case of factoring where a polynomial is in the form a² - b².

Example: For sum of cubes: a³ + b³ → $a+b$ $a² - ab + b²$

Example: For difference of cubes: a³ - b³ → $a-b$ $a² + ab + b²$

Highlight: When factoring quadratics $ax² + bx + c$ , find two factors of ac that add up to b.

Example: Factoring x² + 8x + 12:

Factors of 12 that add up to 8 are 6 and 2
Rewrite as x² + 6x + 2x + 12
Factor by grouping: x $x+6$ + 2 $x+6$
Final result: $x+2$ $x+6$

Vocabulary: Factor by Grouping - A method used to factor polynomials with four or more terms by separating them into groups.

Polynomial Long Division

This section explains the process of polynomial long division, which is similar to regular long division but with polynomials.

Highlight: Don't forget to use placeholders for any missing degrees in the polynomial.

Example: Dividing $2x² + x - 3$ by $x + 1$ :

Divide first terms: 2x² ÷ x = 2x
Multiply: 2x $x + 1$ = 2x² + 2x
Subtract: 2x² + x - 3 - $2x² + 2x$ = -x - 3
Repeat the process with -x - 3
Final result: 2x - 1 with a remainder of -2

Synthetic Division

The guide concludes with an explanation of synthetic division, a shortcut method for dividing polynomials.

Vocabulary: Synthetic Division - A simplified method of polynomial long division, particularly useful when dividing by a linear factor.

Example: Synthetic division of 2x⁴ - 7x³ - 7x + 1 by $x - 4$ :

Write coefficients: 2 -7 0 -7 1
Use 4 as the divisor $root of x - 4$
Bring down first coefficient: 2
Multiply 2 by 4 and add to -7: 2 $4$ + $-7$ = 1
Continue this process for all terms
Final result: 2x³ + x² + 4x + 9 with a remainder of 37

Highlight: Use zero as a placeholder for missing terms in the polynomial during synthetic division.

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

You have the choice, we offer a 100% secure payment via PayPal or an Amazon gift card. There is no limit to how much you can earn.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user