6.4 General Strategy for Factoring Polynomials

Factoring Polynomials

Factoring methods for polynomials

Every factoring problem starts the same way: look for the greatest common factor (GCF) first. After that, the structure of what's left tells you which method to use next.

• Factoring out the GCF
  • Find the largest factor shared by all coefficients and the lowest power of each variable that appears in every term.
  • Divide each term by the GCF and write the results inside parentheses, with the GCF out front.
  • Example: $6x^2 + 9x = 3x(2x + 3)$
• Factoring trinomials where $a = 1$ (form: $x^2 + bx + c$)
  • Find two numbers that multiply to $c$ and add to $b$.
  • Write the factored form as $(x + \text{first number})(x + \text{second number})$.
  • Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$ because $2 \times 3 = 6$ and $2 + 3 = 5$.
• Factoring trinomials where $a \neq 1$ (form: $ax^2 + bx + c$)
  • Multiply $a \times c$ to get the product $ac$.
  • Find two numbers that multiply to $ac$ and add to $b$.
  • Rewrite the middle term using those two numbers, then factor by grouping.
  • Example: $2x^2 + 7x + 3$. Here $ac = 6$, and the pair $1$ and $6$ multiply to $6$ and add to $7$. Rewrite as $2x^2 + x + 6x + 3$, then group: $x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)$.
• Factoring by grouping (four-term polynomials)
  • Group the polynomial into two pairs of terms.
  • Factor the GCF out of each pair.
  • If both groups share a common binomial factor, factor it out.
  • Example: $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$

Complete factorization techniques

The goal is to factor until every piece is irreducible (can't be broken down any further). Here's the general strategy, step by step:

1. Factor out the GCF of the entire polynomial. Always do this first.

2. Count the terms in what remains:

   • Two terms: Check for difference of squares, difference of cubes, or sum of cubes.
   • Three terms: Try trinomial factoring ($a = 1$ or $a \neq 1$ method).
   • Four or more terms: Try factoring by grouping.

3. Check each factor you produced. If any of them can be factored further, keep going.

4. Write the final answer as a product of all irreducible factors.

Example showing the full process:

$12x^2 + 14x - 6$

1. GCF is $2$: $2(6x^2 + 7x - 3)$

2. The trinomial has $a \neq 1$, so use the $ac$ method: $ac = -18$, and the pair $9$ and $-2$ works ($9 \times -2 = -18$, $9 + (-2) = 7$). Factor to get $2(3x - 1)(2x + 3)$.

3. Neither binomial factors further, so the answer is $2(3x - 1)(2x + 3)$.

Special cases in polynomial factoring

Recognizing these patterns saves a lot of time. Train yourself to spot them before you start trial-and-error.

• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$
  • Example: $x^2 - 49 = (x + 7)(x - 7)$
  • Note: The sum of squares $a^2 + b^2$ does not factor over the real numbers.
• Perfect square trinomials: These come from squaring a binomial.
  • $a^2 + 2ab + b^2 = (a + b)^2$
  • $a^2 - 2ab + b^2 = (a - b)^2$
  • How to check: Is the first term a perfect square? Is the last term a perfect square? Does the middle term equal $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$?
  • Example: $x^2 + 6x + 9 = (x + 3)^2$ because $2 \cdot x \cdot 3 = 6x$.
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
  • Example: $8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)(4x^2 + 6x + 9)$
• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
  • Example: $x^3 + 8 = x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)$

Understanding Polynomial Factorization

A few key definitions to keep straight:

• A polynomial is an expression made of variables and coefficients combined using addition, subtraction, and multiplication (no division by variables).
• Factorization means rewriting a polynomial as a product of simpler expressions that multiply back to the original.
• A binomial has exactly two terms (like $3x + 5$), and a trinomial has exactly three.
• Irreducible factors are factors that can't be broken down any further using real numbers. Your final answer should contain only irreducible factors.