Add, subtract, and multiply polynomial expressions with rational number coefficients, recognizing that polynomials are closed under these operations (MA.912.AR.1.1).

What this topic is asking

MA.912.AR.1.1 asks you to add, subtract, and multiply polynomials with rational coefficients, and to recognize that the result is always another polynomial (the closure property). On the B.E.S.T. Algebra 1 EOC these appear as equation-editor items (type the simplified polynomial in standard form) and as multiple choice with sign-error distractors.

Adding and subtracting

Only like terms combine: $5x^2$ and $-2x^2$ combine to $3x^2$, but $5x^2$ and $5x$ do not. The danger is subtraction, because the minus sign in front of a parenthesis distributes to every term inside.

Multiplying

Multiplication uses the distributive property. For a monomial times a polynomial, distribute: $3x(x^2 - 2x + 4) = 3x^3 - 6x^2 + 12x$. For two binomials, every term of the first multiplies every term of the second (FOIL is just this organized):

Watch the middle term: combine the two cross-products carefully, tracking signs.

Closure: why the answer is always a polynomial

The standard highlights closure: when you add, subtract, or multiply polynomials, the result is always another polynomial. Adding or subtracting only combines existing terms, and multiplying adds exponents (which stay whole numbers) and multiplies coefficients (which stay rational). No operation here ever produces a variable in a denominator or under a root, so you never leave the polynomial family. This is the same idea as the integers being closed under addition and multiplication but not division.

How the B.E.S.T. EOC examines this topic

• Equation editor. Simplify a sum, difference, or product and type the polynomial in standard form.
• Multiple choice. Identify a product or simplified expression, with middle-term and sign distractors.
• Multiselect. Select all expressions equivalent to a given polynomial.

A clarifying idea: subtraction is just adding the opposite of every term, so $(P) - (Q)$ is $(P) + (-Q)$. Rewriting the subtraction as an addition of the negated polynomial removes the single most common mistake, the un-distributed minus sign.

Connecting multiplication to area

Multiplying two binomials has a geometric picture that the EOC sometimes uses. The product $(x + 4)(x + 3)$ is the area of a rectangle with sides $x + 4$ and $x + 3$, split into four smaller rectangles with areas $x^2$, $3x$, $4x$, and $12$. Adding those areas gives $x^2 + 7x + 12$, which is exactly the FOIL result. This area model explains why the middle term is the sum of the two cross-products and why no term can be dropped: each piece of the rectangle is a separate region. It also previews factoring, where you run the picture backward from area to side lengths.

Standard form and degree

When you report a polynomial, write it in standard form: terms in descending order of exponent, like $2x^2 - 9x + 9$. The degree is the highest exponent (here $2$), and the leading coefficient is the number on that highest-degree term (here $2$). Standard form matters on the EOC for two reasons. First, equation-editor items are often scored by exact match, so a correct but reordered answer may not match the key, write the largest power first. Second, the degree tells you the function family: degree $1$ is linear, degree $2$ is quadratic, which sets up everything in the Functions and Modeling category. A quick habit is to scan your final answer left to right and confirm the exponents only decrease; if a term is out of order, swap it before you submit.

Why the distributive property is the one rule

Adding, subtracting, and multiplying polynomials all reduce to a single property, the distributive property $a(b + c) = ab + ac$, applied carefully. Addition and subtraction use it in reverse: $3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2$ factors the common $x^2$ out of like terms. Multiplication uses it forward, and FOIL is just the distributive property applied twice, once to break $(x + 4)(x + 3)$ into $x(x + 3) + 4(x + 3)$, then again inside each piece. Seeing every operation as the same property is useful because it means there is really only one thing to get right, distribute completely and track signs, rather than three separate procedures to memorize.

Try this

Q1. Simplify $(5x^2 - 2x + 3) + (-3x^2 + 7x - 8)$. [1 point]

• Cue. $2x^2 + 5x - 5$.

Q2. Expand $(3x - 2)(2x + 5)$. [2 points]

• Cue. $6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10$.