Add, subtract, and multiply polynomial expressions in one variable, including multiplying binomials and applying special products (A.EO.4).

What this topic is asking

A.EO.4 asks you to add, subtract, and multiply polynomials in one variable. On the Virginia Algebra I SOL these are core Expressions and Operations items: combine two polynomials, multiply two binomials, or apply a special product. They appear as multiple choice, fill-in-the-blank (type the standard-form result), and drag-and-drop (build the product term by term).

Adding and subtracting polynomials

A polynomial is a sum of terms, each a number times a variable raised to a whole-number power. To add, combine like terms. To subtract, first distribute the subtraction to every term of the polynomial being subtracted, then combine.

The single biggest subtraction error is sign distribution. $(5x^2 - 2x) - (3x^2 + 4x - 7)$ becomes $5x^2 - 2x - 3x^2 - 4x + 7 = 2x^2 - 6x + 7$: the minus flips every sign inside the second parentheses.

Multiplying a monomial by a polynomial

Distribute the monomial to each term, multiplying coefficients and adding exponents on like bases:

Multiplying two binomials (FOIL)

To multiply $(a + b)(c + d)$, distribute every term of the first across the second. The shortcut FOIL names the four products:

• First: $a \cdot c$
• Outer: $a \cdot d$
• Inner: $b \cdot c$
• Last: $b \cdot d$

Then combine the Outer and Inner terms, which are usually like terms.

The special products

Two patterns appear so often that recognizing them saves time, and they preview the factoring topic:

• Difference of squares. $(a + b)(a - b) = a^2 - b^2$. The middle terms cancel: $(x + 7)(x - 7) = x^2 - 49$.
• Perfect-square trinomial. $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$. So $(x + 5)^2 = x^2 + 10x + 25$.

Why $(a + b)^2$ is not $a^2 + b^2$

The most common multiplication mistake is writing $(x + 3)^2 = x^2 + 9$. Squaring a binomial means multiplying it by itself, $(x + 3)(x + 3)$, which by FOIL gives $x^2 + 3x + 3x + 9 = x^2 + 6x + 9$. The missing $6x$ is the sum of the Outer and Inner products, $2ab$. An area picture makes it concrete: a square of side $(x + 3)$ has area $(x + 3)^2$, and it splits into a big square $x^2$, a small square $9$, and two rectangles each $3x$. Forgetting the two rectangles is forgetting the $2ab$ term. The exponent does not distribute across a sum, which is the same reason $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$.

How the SOL examines this topic

• Fill-in-the-blank. Multiply or combine polynomials and type the standard-form result.
• Multiple choice. Pick the correct product or sum, with distractors built from sign and middle-term errors.
• Drag-and-drop. Assemble a product term by term, or order the steps of a multiplication.

Try this

Q1. Simplify $(3x^2 + x) + (x^2 - 4x + 2)$. [1 point]

• Cue. $4x^2 - 3x + 2$.

Q2. Expand $(x + 6)(x - 2)$. [2 points]

• Cue. $x^2 - 2x + 6x - 12 = x^2 + 4x - 12$.