Add, subtract, and multiply polynomials, and factor completely using the greatest common factor, the difference of two squares, and trinomial factoring.

What this topic is asking

The Algebra category requires fluent polynomial arithmetic (the A-APR standards) and complete factoring. On the Grade 10 MCAS you add, subtract, and multiply polynomials, and you factor expressions completely. Factoring underpins solving quadratics and finding zeros, so it is one of the highest-leverage skills on the test, and much of it falls in the no-calculator session where you must do it by hand.

Adding and subtracting polynomials

Combine like terms, those with the same variable and exponent. The only trap is the subtraction sign, which must be distributed across every term in the parentheses that follows it.

Notice every sign inside the second parentheses flipped: $-x$ became $+x$ and $+3$ became $-3$. Forgetting to distribute the minus across the later terms is the single most common error here.

Multiplying polynomials

Multiplication is the distributive property applied to every pair of terms. For two binomials, FOIL (First, Outer, Inner, Last) is a checklist:

For a binomial times a trinomial, distribute each term of the binomial across the whole trinomial, then combine like terms. Two special products are worth memorizing because they appear often and connect to factoring:

The square of a binomial has a middle term $2ab$; a frequent error is writing $(x + 3)^2 = x^2 + 9$ and dropping the $6x$.

Factoring completely, in order

"Factor completely" means break the polynomial into factors that cannot be factored further. Work in a fixed order so nothing is missed.

1. Greatest common factor (GCF). Pull out the largest factor common to every term. For $6x^3 + 9x^2$, the GCF is $3x^2$: $3x^2(2x + 3)$.
2. Difference of two squares. $a^2 - b^2 = (a - b)(a + b)$. So $x^2 - 25 = (x - 5)(x + 5)$.
3. Trinomial factoring. For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. For $x^2 + 7x + 12$, the numbers $3$ and $4$ work: $(x + 3)(x + 4)$. For $ax^2 + bx + c$ with $a \neq 1$, find two numbers multiplying to $ac$ and adding to $b$, split the middle term, and factor by grouping.

Recognizing the AC method

When the leading coefficient is not 1, the AC method keeps factoring reliable. To factor $2x^2 + 7x + 3$: multiply $a \cdot c = 2 \cdot 3 = 6$, find two numbers with product 6 and sum 7 (namely 6 and 1), split the middle term as $2x^2 + 6x + x + 3$, and group: $2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$. Checking by expanding confirms the result, which is a good no-calculator habit.

The same grouping idea handles a four-term polynomial directly. For $x^3 + 2x^2 + 3x + 6$, group in pairs: $x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3)$. Grouping works when the two pairs share a common binomial factor after each pair's GCF is removed; if they do not match, try regrouping the terms in a different order.

Why factoring matters and how the MCAS uses it

Factoring is not an end in itself on the Grade 10 MCAS; it is the gateway to solving and to finding zeros. A factored quadratic such as $(x - 3)(x + 5)$ immediately gives the zeros $x = 3$ and $x = -5$ through the zero-product property, and those zeros are the $x$-intercepts of the matching parabola. So a single factoring step can answer a question about solutions, intercepts, or where a quantity equals zero.

Multiple-select items sometimes give several factored expressions and ask which are equivalent to a given polynomial, or which share a particular factor. Being fluent in both directions, expanding to check and factoring to reveal, lets you verify a choice quickly. A reliable check is to expand your factors and confirm you recover the original polynomial, term for term, including the middle term.

Try this

Q1. Expand $(x - 5)^2$.

• Cue. $x^2 - 10x + 25$ (do not forget the middle term).

Q2. Factor $x^2 - 3x - 10$.

• Cue. $(x - 5)(x + 2)$.