8.2 Factoring Polynomial Expressions
Key Takeaways
- MA.912.AR.1.7 covers factoring out the GCF, factoring trinomials x^2+bx+c and ax^2+bx+c, the difference of squares a^2-b^2=(a+b)(a-b), perfect square trinomials a^2+-2ab+b^2=(a+-b)^2, factoring by grouping, and recognizing that a sum of squares a^2+b^2 is not factorable over the reals
- Always factor out the GCF first before applying any other factoring pattern; failing to do so leaves an answer that is technically correct but incomplete and will not match EOC answer choices
- For trinomials x^2+bx+c, find two numbers that multiply to c and add to b; when c is positive both factors share the sign of b, and when c is negative the factors have opposite signs with the larger taking the sign of b
- For ax^2+bx+c with a>1, use the AC method: multiply a*c, find two numbers that multiply to ac and add to b, split the middle term, then factor by grouping
- A sum of squares such as x^2+9 or 4x^2+25 cannot be factored over the real numbers; (a+b)^2 equals a^2+2ab+b^2, which includes a middle cross term that the sum of squares lacks
Quick Answer: MA.912.AR.1.7 requires factoring polynomial expressions using the greatest common factor (GCF), factoring trinomials x^2 + bx + c and ax^2 + bx + c, the difference of squares a^2 - b^2 = (a + b)(a - b), perfect square trinomials a^2 +- 2ab + b^2 = (a +- b)^2, factoring by grouping, and knowing that a sum of squares a^2 + b^2 is not factorable over the reals.
Factoring Out the GCF
Always begin by checking for a greatest common factor across all terms. The GCF is the largest monomial (coefficient and variable powers) that divides every term. Factoring the GCF first is mandatory because EOC answer choices are fully factored; 2x^2 + 8x + 8 written as (x + 2)(x + 4) is incomplete and will not match the keyed choice 2(x + 2)^2.
Worked example. Factor 12x^3 + 18x^2. GCF of coefficients 12 and 18 is 6. GCF of x^3 and x^2 is x^2 (the lowest power). So GCF = 6x^2. Divide each term: 12x^3 / 6x^2 = 2x, 18x^2 / 6x^2 = 3. Result: 6x^2(2x + 3).
Factoring Trinomials x^2 + bx + c
For a trinomial with leading coefficient 1, find two numbers that multiply to c and add to b.
Worked example. Factor x^2 + 7x + 12. Need two numbers multiplying to 12 and adding to 7. Pairs whose product is 12: (1, 12) sum 13, (2, 6) sum 8, (3, 4) sum 7. So x^2 + 7x + 12 = (x + 3)(x + 4).
Worked example with negative c. Factor x^2 + 2x - 8. Need product -8 and sum +2. One positive, one negative; the larger factor takes the sign of b. (4)(-2) = -8, sum 2. Result: (x + 4)(x - 2). When c is positive, factors share the sign of b; when c is negative, factors have opposite signs.
Factoring Trinomials ax^2 + bx + c (a > 1)
Use the AC method: multiply a * c, find two numbers that multiply to ac and add to b, split the middle term, then factor by grouping.
Worked example. Factor 2x^2 + 7x + 3. a * c = 6. Find two numbers multiplying to 6 and adding to 7: 6 and 1. Split: 2x^2 + 6x + 1x + 3. Group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Difference of Squares
A difference of squares a^2 - b^2 factors as (a + b)(a - b). Recognize it: two terms, both perfect squares, separated by subtraction. Factor 9x^2 - 25: (3x)^2 - 5^2 = (3x + 5)(3x - 5). With two variables, 16x^2 - 49y^2 = (4x + 7y)(4x - 7y).
Perfect Square Trinomials
A perfect square trinomial has the form a^2 +- 2ab + b^2 = (a +- b)^2. Check that the first and last terms are perfect squares and the middle term is exactly twice the product of their roots. Factor x^2 + 10x + 25: x^2 is a square, 25 = 5^2, and 10 = 2 * x * 5, so the result is (x + 5)^2. With a negative middle, 4x^2 - 12x + 9 = (2x - 3)^2.
Factoring by Grouping
Used for four-term polynomials or as a step in the AC method. Group terms in pairs, factor the GCF from each pair, then factor out the common binomial. Factor x^3 + 2x^2 + 3x + 6: group (x^3 + 2x^2) + (3x + 6) = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2).
Sum of Squares: Not Factorable
A sum of squares a^2 + b^2 cannot be factored over the real numbers. EOC distractors offer (a + b)^2, but that equals a^2 + 2ab + b^2, which includes the cross term 2ab that the sum of squares lacks. Mark x^2 + 9, 4x^2 + 25, or x^2 + 16y^2 as prime.
Common EOC Traps
- Forgetting to factor the GCF first. 2x^2 + 8x + 8 should be 2(x^2 + 4x + 4) = 2(x + 2)^2, not jumped straight to a trinomial pattern that ignores the leading 2.
- Treating a sum of squares as factorable. x^2 + 16 is prime. It is not (x + 4)^2 (that gives x^2 + 8x + 16) and not (x + 4)(x - 4) (that gives x^2 - 16).
- Sign errors with negative c. When c is negative, the factors have opposite signs; the larger takes the sign of b. For x^2 + 2x - 8, the correct factors are (x + 4)(x - 2), not (x - 4)(x + 2).
- Stopping at one level of factoring. 3x^2 - 12 = 3(x^2 - 4) = 3(x + 2)(x - 2). EOC answer choices are fully factored.
Factoring Decision Flow
| Polynomial form | First strategy |
|---|---|
| Any polynomial | Check GCF first |
| 4 terms | Group pairs |
| 3 terms, a = 1 | Find factors of c summing to b |
| 3 terms, a > 1 | AC method, then grouping |
| 2 terms, both squares, subtraction | Difference of squares |
| 2 terms, both squares, addition | Not factorable (prime) |
Connection to Quadratic Solving
Factoring is the gateway to solving quadratic equations by the zero-product property in MA.912.AR.3.4 (chapter 9). Once a quadratic is factored as (x - r1)(x - r2), setting each factor to zero gives the roots. EOC items frequently chain a factoring step into a solving step, so master the decision flow above to identify the correct strategy in under 30 seconds on test day.
Factor 2x^2 + 7x + 3 completely.
Which expression is fully factored and equivalent to 3x^2 - 12?
Which of the following polynomial expressions is NOT factorable over the real numbers?
Factor x^3 + 2x^2 + 3x + 6 by grouping.