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.