Polynomials and Factoring
Key Takeaways
- Polynomials are sums of monomials with nonnegative integer exponents; degree is the highest exponent.
- Add or subtract polynomials by combining like terms; multiply using distribution or area models.
- Factor out the GCF first; then use difference of squares, trinomial patterns, or grouping.
- The Factor Theorem links linear factors to roots: (x − r) is a factor iff f(r) = 0.
- Praxis items often ask which factored form is equivalent or which step exposes a student error.
Polynomials as the Language of Algebra
On Praxis 5165, polynomials appear in simplification, factoring, solving equations, and previewing function behavior. The official outline lists polynomial structure explicitly. You need operational fluency and pattern recognition—not abstract ring theory.
A polynomial in x is a sum a_n x^n + … + a_1 x + a_0 with integer exponents n ≥ 0. The degree is the largest exponent with nonzero coefficient.
Operations
Add/subtract: combine like terms only.
(3x^2 − 5x + 1) + (x^2 + 2x − 4) = 4x^2 − 3x − 3.
Multiply: distribute each term.
Worked Example 1
(x − 3)(x^2 + 2x + 4) = x^3 + 2x^2 + 4x − 3x^2 − 6x − 12 = x^3 − x^2 − 2x − 12.
Special products to memorize:
| Pattern | Expansion |
|---|---|
| (a + b)^2 | a^2 + 2ab + b^2 |
| (a − b)^2 | a^2 − 2ab + b^2 |
| (a + b)(a − b) | a^2 − b^2 |
Worked Example 2
(2x − 5)^2 = 4x^2 − 20x + 25. Missing the middle term is a top student error.
Factoring Strategy
- Greatest common factor (GCF) always first.
- Count terms: two terms → difference of squares; three terms → trinomial; four terms → grouping.
Difference of squares: a^2 − b^2 = (a + b)(a − b).
Worked Example 3
Factor 9x^2 − 16 = (3x)^2 − (4)^2 = (3x − 4)(3x + 4).
Worked Example 4 — GCF first
4x^3 − 12x^2 = 4x^2(x − 3). Without factoring 4x^2, trinomial methods fail.
Trinomial x^2 + bx + c: find integers m, n with mn = c and m + n = b.
Worked Example 5
x^2 − 5x + 6 = (x − 2)(x − 3).
For ax^2 + bx + c when a ≠ 1, use AC splitting or trial grouping.
Worked Example 6
6x^2 + 7x − 3: AC = −18. Split 7 into 9 and −2.
6x^2 + 9x − 2x − 3 = 3x(2x + 3) − 1(2x + 3) = (3x − 1)(2x + 3).
Grouping (four terms):
xy + 2y + 3x + 6 = y(x + 2) + 3(x + 2) = (x + 2)(y + 3).
Solving by Factoring
Set polynomial equal to zero; use the zero product property.
Worked Example 7
x^2 − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or 3.
Worked Example 8
2x^2 − 8x = 0 → 2x(x − 4) = 0 → x = 0 or 4.
Factor Theorem Connection
For polynomial f(x), (x − r) is a factor if and only if f(r) = 0. If x − 2 is a factor of f, then f(2) = 0. This bridges to rational roots and graph intercepts in later chapters.
Worked Example 9
If f(x) = x^3 − 4x, then f(2) = 8 − 8 = 0, so (x − 2) is a factor. Indeed f(x) = x(x − 2)(x + 2).
Common Student Errors (Teaching Items)
- Forgetting to factor out a GCF, making trinomial factoring impossible.
- Sign errors in (a − b)^2, writing a^2 − b^2 instead of a^2 − 2ab + b^2.
- Confusing factor and term cancellation in rational expressions (preview of next section).
When reviewing student work, expand the proposed factors to verify equivalence.
Exam Strategy
If multiple-choice answers are in factored form, multiply back mentally. If asked for degree, identify the highest exponent only after full simplification. Polynomial long division appears less often than factoring, but (x^3 − 1)/(x − 1) = x^2 + x + 1 for x ≠ 1 is fair game.
Polynomial Degree and Leading Coefficient
For f(x) = −3x^4 + 2x − 7, degree is 4 and leading coefficient is −3. Degree determines end behavior on graphs—a bridge to the functions chapter.
Multiplying Three Binomials
(x + 1)(x − 2)(x + 3): multiply two first, then distribute.
(x + 1)(x − 2) = x^2 − x − 2. Then (x^2 − x − 2)(x + 3) = x^3 + 2x^2 − 5x − 6.
Factoring Completely
Worked Example 10
2x^3 − 18x = 2x(x^2 − 9) = 2x(x − 3)(x + 3).
"Factor completely" means no further factoring is possible over the integers.
Remainder Intuition
When dividing by (x − c), the remainder equals f(c). If f(1) = 0, then (x − 1) is a factor. Quick evaluation saves time on multiple-choice factor questions.
Evaluating Polynomials Efficiently
For f(x) = x^3 − 2x + 5, f(−2) = (−8) − (−4) + 5 = 1. Synthetic substitution is optional; direct substitution is fine on Praxis if you track signs.
Perfect Square Trinomials
Recognize a^2 + 2ab + b^2 = (a + b)^2. Example: x^2 + 6x + 9 = (x + 3)^2. Mis-factoring as (x + 3)(x − 3) confuses sum and difference patterns.
Zeros and x-Intercepts
If (x − 5) is a factor, x = 5 is a zero. Multiple-choice questions may ask for the sum of zeros after factoring—add the roots from linear factors.
Which factorization is correct for x^2 − 9?
What are the solutions to x^2 − 5x + 6 = 0?