Add, subtract, and multiply polynomials, and factor quadratic expressions including GCF, trinomials, and the difference of squares (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard covers the core polynomial fluency the rest of the course leans on: adding, subtracting, and multiplying polynomials, and factoring quadratic expressions back into products. Factoring is the higher-leverage half, because it is the engine for solving quadratics by the zero-product property later in the course. On the Georgia Milestones EOC these appear as one-point selected-response items (a single product or factorization) and as two-point constructed-response items that ask for complete factoring with reasoning shown.

Adding and subtracting polynomials

To add or subtract, combine like terms, terms with the same variable raised to the same power. Subtraction needs care: distribute the negative sign across every term in the second polynomial.

Notice that $-(- 4x) = +4x$ and $-(+6) = -6$. The single most common error here is subtracting only the first term of the second polynomial and forgetting the rest.

Multiplying polynomials

Multiplication uses the distributive property: every term of one factor multiplies every term of the other. For two binomials the shortcut is FOIL (First, Outer, Inner, Last).

A useful special case is the square of a binomial: $(x + a)^2 = x^2 + 2ax + a^2$, which is the perfect-square-trinomial pattern you will reverse when completing the square.

Factoring: a fixed order

Factoring is multiplication run backward. Work in a reliable order so you do not miss a step.

1. GCF first. Factor out the greatest common factor of all terms: $6x^2 + 9x = 3x(2x + 3)$.
2. Difference of squares. $a^2 - b^2 = (a - b)(a + b)$. So $9x^2 - 16 = (3x - 4)(3x + 4)$, taking $a = 3x$, $b = 4$.
3. Trinomial. For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$: $x^2 - 7x + 12 = (x - 3)(x - 4)$ because $-3$ and $-4$ multiply to $12$ and add to $-7$.

How the Milestones examines this topic

• Multiple choice. A single product or a single factorization, with sign-error and middle-term distractors.
• Numeric or drag-and-drop entry. Build a product or assemble factors from a set of binomials.
• Constructed response. Factor completely (often a trinomial and a difference of squares) with reasoning shown; partial credit for one correct factorization.

Why "multiply to c, add to b" works

The trinomial method is not a trick; it falls straight out of FOIL. When you expand $(x + a)(x + b)$ you get $x^2 + (a + b)x + ab$, so the constant term is the product $ab$ and the middle coefficient is the sum $a + b$. Factoring just reverses this: you are searching for the two numbers $a$ and $b$ whose product is the constant and whose sum is the middle coefficient. Listing the factor pairs of the constant and checking which pair adds to the middle coefficient is therefore a guaranteed method, and seeing the connection to FOIL is what lets you handle the sign cases (both negative, mixed signs) without guessing.

Factoring with a leading coefficient

When the leading coefficient is not 1, as in $2x^2 + 7x + 3$, first check for a GCF (here there is none), then either test binomial factor pairs or use the grouping (AC) method: multiply $a \cdot c = 2 \cdot 3 = 6$, find two numbers that multiply to 6 and add to 7 (namely 6 and 1), split the middle term as $2x^2 + 6x + x + 3$, and group: $2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)$. Grouping turns a harder trinomial into the same "two numbers" search, which is why the EOC expects you to handle leading coefficients other than 1 by this systematic route rather than by trial and error.

Try this

Q1. Expand $(x - 6)(x + 6)$. [1 point]

• Cue. Difference of squares: $x^2 - 36$.

Q2. Factor $x^2 + 9x + 20$ completely. [1 point]

• Cue. Two numbers multiplying to 20 and adding to 9 are 4 and 5: $(x + 4)(x + 5)$.