Factoring Polynomials

Factoring Polynomials

Introduction

• Continuing work with factorization, focusing on polynomials.
• Factorization of polynomials is crucial for Math 107.
• Various methods of factorization will be explored thoroughly.

Objectives

• Easy Factorization:
  • Identifying common factors in all terms of the polynomial.
  • Finding the greatest common monomial factor.
    • Monomial: One term (e.g., 3x^2).
• Grouping Terms:
  • Special factorization involving a factor (monomial, binomial, trinomial, etc.) common to all terms.
• Special Products:
  • Applying special product formulas for factorization.
• Factoring Completely:
  • Ensuring the polynomial is factored as much as possible.

Finding the Greatest Common Monomial Factor

• Example Polynomial: 3x^5 + 6x^3 - 9x
• Degree of polynomial: 5
• Terms are in descending order of power (5, 3, 1); powers 4 and 2, and the constant are missing.
• Leading coefficient: 3
• Factor out x because it's common to all terms.
  • x^5 = x^4 * x
  • x^3 = x^2 * x
  • x = x * 1

Identifying Common Coefficients

• Coefficients are 3, 6, and -9.
• Factorize each coefficient:
  • 3 = 3 * 1
  • 6 = 3 * 2
  • -9 = 3 * -3
• Common factor: 3

Determining the Monomial Factor

• Monomial factor: Product of common coefficient and common variable.
• In this case, 3x.
• General form of a monomial factor: a * x^n, where a is a real number.
• Because it's common, it is termed the greatest common monomial factor.

More Examples of Finding the Greatest Common Monomial Factor

• Example 1: 24x^3 - 32x^2
  • Coefficients: 24 and -32
  • Factorization: 24 = 8 * 3, -32 = 8 * -4
  • Common factor: 8
  • Variables: x^3 and x^2
  • Factorization: x^3 = x^2 * x, x^2 = x^2 * 1
  • Common factor: x^2
  • Greatest common monomial factor: 8x^2
  • Resulting factorization: 8x^2(3x - 4)
• Example 2: 24x^5 - 21x^3 + 6x^2
  • Coefficients: 24, -21, and 6
  • Factorization: 24 = 3 * 8, -21 = 3 * -7, 6 = 3 * 2
  • Common factor: 3
  • Variables: x^5, x^3, and x^2
  • Factorization: x^5 = x^2 * x^3, x^3 = x^2 * x, x^2 = x^2 * 1
  • Common factor: x^2
  • Greatest common monomial factor: 3x^2
  • Resulting factorization: 3x^2(8x^3 - 7x + 2)

Common Binomial Factors

• Consider expressions like 5x^2(6x - 5) - 2(6x - 5).
• (6x - 5) is a common binomial factor.
• Factor out the common binomial: (6x - 5)(5x^2 - 2).
• This also applies to trinomial or polynomial factors.
• Check if the remaining expression can be factored further.

Example with Common Binomial Factors

• Problem: x^2(x - 3) + 7(x - 3)
• Common binomial factor: (x - 3)
• Factorization: (x - 3)(x^2 + 7)

Factorization by Grouping

• Problem: x^3 - 3x^2 + 7x - 21
• Degree of polynomial: 3
• Leading term: x^3
• Constant term: -21
• Group terms: (x^3 - 3x^2) + (7x - 21)
• Factor each group:
  • x^3 - 3x^2 = x^2(x - 3)
  • 7x - 21 = 7(x - 3)
• Rewrite the expression: x^2(x - 3) + 7(x - 3)
• Factor out the common binomial factor: (x - 3)(x^2 + 7)
• Convenient grouping: Choosing terms to facilitate finding a monomial factor.

Special Factors: Difference of Two Squares

• Formula: u^2 - v^2 = (u + v)(u - v)
• Both terms must be perfect squares.
• Crucially, there must be a difference (subtraction) between the terms.
• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 81, etc.
• Example: x^2 - 4
  • x^2 is a perfect square.
  • 4 is a perfect square (2^2).
  • u = x, v = 2
  • Factorization: (x + 2)(x - 2)

More Complex Example of Difference of Two Squares

• Problem: 25x^2 - 49y^2
• 25 is a perfect square (5^2).
• 49 is a perfect square (7^2).
• x^2 and y^2 are perfect squares.
• u = 5x, v = 7y
• Factorization: (5x + 7y)(5x - 7y)
  Note: a plus sign between the two perfect squares makes this factorization impossible!!

Further Examples

• 4y^2 - 36x^2 = (2y + 6x)(2y - 6x)
• 81x^2 - 4 = (9x + 2)(9x - 2)
• Terms must be perfect squares, and there must be a difference.

Special Factors: Sum and Difference of Cubes

• Sum of Cubes: u^3 + v^3 = (u + v)(u^2 - uv + v^2)
• Difference of Cubes: u^3 - v^3 = (u - v)(u^2 + uv + v^2)
• Remembering perfect cubes is important.
• Perfect Cubes: 1, 8, 27, 64, 125, 216, etc.

Examples of Sum and Difference of Cubes

• Example 1: x^3 - 64
  • x^3 is a perfect cube.
  • 64 is a perfect cube (4^3).
  • u = x, v = 4
  • Using the difference of cubes formula:
    • (x - 4)(x^2 + 4x + 16)
• Example 2: 8x^3 + 125
  • 8 is a perfect cube (2^3).
  • 125 is a perfect cube (5^3).
  • u = 2x, v = 5
  • Using the sum of cubes formula:
    • (2x + 5)(4x^2 - 10x + 25)

Practice Problem

• Problem: y^3 - 27x^3
  • u = y, v = 3x
  • Using the difference of cubes formula:
  • (y - 3x)(y^2 + 3xy + 9x^2)
  • Use alphabetic order for variables (e.g., xy instead of yx).

Summary

• Special Formulas:
  • Difference of Squares: u^2 - v^2 = (u + v)(u - v)
  • Sum of Cubes: u^3 + v^3 = (u + v)(u^2 - uv + v^2)
  • Difference of Cubes: u^3 - v^3 = (u - v)(u^2 + uv + v^2)
• Factorization by grouping (common binomial factors).
• Common monomial factors.
• Tomorrow: Factorization of trinomials.