Add, subtract, and multiply polynomials (closure), and factor quadratic and higher expressions using common factors, the difference of two squares, and trinomial factoring, including a leading coefficient other than 1.

What this topic is asking

The Regents Algebra I exam (the Arithmetic with Polynomials, A-APR, and Seeing Structure in Expressions, A-SSE, clusters) expects you to add, subtract, and multiply polynomials fluently, and to factor them: pulling out a common factor, recognizing a difference of two squares, and factoring trinomials whether the leading coefficient is $1$ or not. These skills underpin solving quadratics and analyzing graphs, so they recur all over the exam.

Adding, subtracting, and multiplying

Like terms have the same variable raised to the same power. Adding and subtracting polynomials means combining like terms, and the only common error is the sign.

Notice the subtraction sign flips every term of the second polynomial. Multiplication uses the distributive property: every term of one factor multiplies every term of the other.

Because adding, subtracting, or multiplying polynomials always gives another polynomial, we say polynomials are closed under these operations, a fact the Regents sometimes asks about directly.

Factoring: the standard order

Factoring works best in a fixed order, and the first step is always the same.

Factoring a trinomial with leading coefficient 1

When $a = 1$, the trinomial $x^2 + bx + c$ factors as $(x + p)(x + q)$, where $p$ and $q$ multiply to $c$ and add to $b$. For $x^2 - 7x + 12$, you need two numbers with product $12$ and sum $-7$: those are $-3$ and $-4$, so $x^2 - 7x + 12 = (x - 3)(x - 4)$. The signs follow a pattern: a positive $c$ with a negative $b$ means both numbers are negative; a negative $c$ means the numbers have opposite signs.

A clarifying point is that "factor completely" is a single instruction with several steps. After the first factor comes out, you must re-examine what is left. In $3x^2 - 27$, pulling out $3$ gives $3(x^2 - 9)$, but $x^2 - 9$ is still a difference of squares, so the complete answer is $3(x - 3)(x + 3)$. The Regents reserves full credit for the fully factored form, and expanding your answer to confirm it equals the original is the surest check.

Try this

Q1. Multiply $(x + 5)(x - 5)$. [1 credit]

• Cue. Difference of squares pattern: $x^2 - 25$.

Q2. Factor $x^2 + 2x - 15$. [2 credits]

• Cue. Two numbers multiply to $-15$ and add to $2$: that is $5$ and $-3$, so $(x + 5)(x - 3)$.