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Easy Steps: How to Factor Polynomials and Solve Problems

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Easy Steps: How to Factor Polynomials and Solve Problems

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 division.

Overall Summary:

A comprehensive guide on how to factor polynomials and perform polynomial division, covering:

  • Factoring techniques for polynomials with different numbers of terms
  • Polynomial long division
  • Synthetic division
  • Examples and step-by-step solutions for various factoring and division problems

6/12/2023

367

(
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:

  1. Factoring out the Greatest Common Factor (GCF)
  2. Difference of Squares
  3. Sum or Difference of Cubes
  4. Factoring Quadratics
  5. Factoring by Grouping
  6. 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:

  1. Factors of 12 that add up to 8 are 6 and 2
  2. Rewrite as x² + 6x + 2x + 12
  3. Factor by grouping: x(x+6) + 2(x+6)
  4. 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):

  1. Divide first terms: 2x² ÷ x = 2x
  2. Multiply: 2x(x + 1) = 2x² + 2x
  3. Subtract: 2x² + x - 3 - (2x² + 2x) = -x - 3
  4. Repeat the process with -x - 3
  5. 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):

  1. Write coefficients: 2 -7 0 -7 1
  2. Use 4 as the divisor (root of x - 4)
  3. Bring down first coefficient: 2
  4. Multiply 2 by 4 and add to -7: 2(4) + (-7) = 1
  5. Continue this process for all terms
  6. 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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Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying

Easy Steps: How to Factor Polynomials and Solve Problems

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 division.

Overall Summary:

A comprehensive guide on how to factor polynomials and perform polynomial division, covering:

  • Factoring techniques for polynomials with different numbers of terms
  • Polynomial long division
  • Synthetic division
  • Examples and step-by-step solutions for various factoring and division problems

6/12/2023

367

 

10th

 

Algebra 2

24

(
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

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Factoring Techniques

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

  1. Factoring out the Greatest Common Factor (GCF)
  2. Difference of Squares
  3. Sum or Difference of Cubes
  4. Factoring Quadratics
  5. Factoring by Grouping
  6. 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:

  1. Factors of 12 that add up to 8 are 6 and 2
  2. Rewrite as x² + 6x + 2x + 12
  3. Factor by grouping: x(x+6) + 2(x+6)
  4. 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):

  1. Divide first terms: 2x² ÷ x = 2x
  2. Multiply: 2x(x + 1) = 2x² + 2x
  3. Subtract: 2x² + x - 3 - (2x² + 2x) = -x - 3
  4. Repeat the process with -x - 3
  5. 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):

  1. Write coefficients: 2 -7 0 -7 1
  2. Use 4 as the divisor (root of x - 4)
  3. Bring down first coefficient: 2
  4. Multiply 2 by 4 and add to -7: 2(4) + (-7) = 1
  5. Continue this process for all terms
  6. 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.

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

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying