Do I have to use FOIL to multiply binomials?

No. FOIL is just a memory aid for the distributive property applied to two binomials, and it stops working the moment a factor has three or more terms. The box (area) method or simply distributing each term of the first factor across the whole second factor works for any sizes and makes it easier to see that you have used every pair of terms.

How do I know if an expression is not a polynomial?

Look at the exponents on the variables. Every exponent must be a whole number: 0, 1, 2, 3,. So x^-2, 7/x, sqrt(x), and x^1/3 all disqualify an expression. Numbers in denominators are fine, though — x/4 is the same as 1/4x, which is a legitimate polynomial term.

Why does my answer have fewer terms than the number of products I found?

Because some of the products turn out to be like terms and merge. Multiplying two binomials produces four products, but the outer and inner products usually share the same variable power, so they combine into a single middle term. That is why (2x - 3)(x + 5) gives the three-term answer 2x^2 + 7x - 15.

What is the fastest way to catch a sign mistake?

Substitute a convenient number such as x = 2 into both the original expression and your simplified answer. If the two values differ, you have an error, and it is almost always a minus sign that failed to reach every term inside a set of parentheses. Avoid using x = 0 or x = 1 alone, since those can hide certain mistakes.

ALG1-7.1

Adding, Subtracting & Multiplying Polynomials

Learn to name polynomial parts, write standard form, and add, subtract, and multiply polynomials — including binomial by binomial — with worked examples and common pitfalls.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Adding, Subtracting & Multiplying Polynomials, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

A polynomial is just a sum of terms built from numbers and whole-number powers of a variable, but that simple definition powers almost everything that comes later in Algebra 1. Before you can factor an expression or solve a quadratic equation, you have to be fluent at building and rearranging polynomials.

In this lesson you will learn the vocabulary — term, coefficient, degree, leading coefficient — and how to put a polynomial in standard form. Then you will combine polynomials three ways: adding, subtracting, and multiplying. Along the way you will see why every one of these operations produces another polynomial, a property called closure. The two places students most often stumble are subtraction (the minus sign has to reach every term) and multiplication (every term in the first factor must meet every term in the second), so we will slow down on both.

Parts of a Polynomial and Standard Form

A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents:

7

x

-4x^3

, and

\frac{2}{3}xy^2

all qualify. A polynomial is a sum of monomials, and each monomial in the sum is called a term.

In the term

-4x^3

, the number

-4

is the coefficient and

3

is the degree of that term. The degree of the whole polynomial is the largest degree among its terms. The term with that largest degree is the leading term, and its coefficient is the leading coefficient. A term with no variable, like

9

, is the constant term and has degree

0

.

Standard form means writing the terms in order from highest degree to lowest. So

5 - 2x^3 + x

becomes

-2x^3 + x + 5

. Notice the sign travels with the term: the leading coefficient here is

-2

, not

2

. Forgetting to carry the sign is the single most common error in this part of the lesson.

Polynomials are also named by how many terms they have and by their degree.

Expression	Terms	Degree	Names
$6x$	1	1	linear monomial
$x^2 - 9$	2	2	quadratic binomial
$3x^2 + x - 5$	3	2	quadratic trinomial
$-x^3 + 4x$	2	3	cubic binomial

One caution:

\frac{4}{x}

and

\sqrt{x}

are not polynomial terms, because

\frac{4}{x} = 4x^{-1}

and

\sqrt{x} = x^{1/2}

do not have whole-number exponents.

Adding and Subtracting: Like Terms Only

Adding polynomials means combining like terms — terms with exactly the same variable and exactly the same exponent. You add the coefficients and keep the variable part unchanged:

(4x^2 + 3x - 1) + (2x^2 - 7x + 5) = 6x^2 - 4x + 4

Here

4x^2

and

2x^2

are like terms,

3x

and

-7x

are like terms, and

-1

and

5

are like terms. But

x^2

and

x

are never like terms, no matter how tempting it looks. Writing

4x^2 + 3x = 7x^3

7x^2

is wrong; those terms simply stay separate.

Subtraction is where most errors happen. The minus sign in front of a parenthesis applies to every term inside, not just the first one. Rewrite subtraction as adding the opposite:

(5x^2 - 2x + 6) - (3x^2 + 4x - 9)

= 5x^2 - 2x + 6 - 3x^2 - 4x + 9

= 2x^2 - 6x + 15

Watch the last term:

-(-9)

became

+9

. A student who distributes the minus sign only to

3x^2

would get

2x^2 + 2x - 3

, which is a completely different polynomial. A reliable habit is to physically rewrite the second polynomial with every sign flipped before you combine anything.

A useful check: substitute a number, say

x = 2

, into the original expression and into your answer. The original gives

(20 - 4 + 6) - (12 + 8 - 9) = 22 - 11 = 11

, and the answer gives

8 - 12 + 15 = 11

. Matching values is strong evidence you simplified correctly.

Multiplying a Monomial by a Polynomial

Multiplying uses the distributive property plus the product rule for exponents,

x^m \cdot x^n = x^{m+n}

. Multiply the monomial by each term of the polynomial, one at a time:

3x^2(4x^3 - 5x + 2) = 12x^5 - 15x^3 + 6x^2

Check each piece. Coefficients multiply:

3 \cdot 4 = 12

. Exponents add:

x^2 \cdot x^3 = x^5

. The middle term is

3x^2 \cdot (-5x) = -15x^3

, and the last is

3x^2 \cdot 2 = 6x^2

— the variable does not disappear just because the other factor is a constant.

Two mistakes show up constantly. First, multiplying the exponents instead of adding them, giving

x^6

instead of

x^5

. Second, forgetting the final term of the polynomial, especially when the expression is long or the last term is negative. Draw arrows from the monomial to each term so you can count that you used every one.

Signs follow the ordinary rules: a negative monomial flips the sign of every product.

-2x(x^2 - 3x + 4) = -2x^3 + 6x^2 - 8x

When more than one variable appears, group like bases:

5ab^2(2a^3b) = 10a^4b^3.

This skill also shows up inside equation solving. If a problem asks you to solve

3(x + 4) - 2x = 20

, you are distributing a monomial exactly the same way; polynomial multiplication is that same move with higher powers.

Multiplying Binomial by Binomial, and Why Closure Matters

To multiply two binomials, every term in the first factor must be multiplied by every term in the second — four products in all. Some people remember this as FOIL (First, Outer, Inner, Last), but it is really just the distributive property used twice, and FOIL only works for two binomials. The box method generalizes to any size.

(2x - 3)(x + 5)

$\times$	$x$	$+5$
$2x$	$2x^2$	$10x$
$-3$	$-3x$	$-15$

Adding the four cells:

2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15

. Notice that the two middle products are like terms, so they combine into one. That is why a binomial times a binomial usually gives a trinomial.

Carry the sign of a term into the box with it. In the row for

-3

, the products are

-3x

and

-15

, not

3x

and

15

. Sign slips in the third row are the leading cause of wrong middle terms.

The big idea behind all of this is closure: add, subtract, or multiply two polynomials and the result is always another polynomial. The reason is structural — adding coefficients keeps them as numbers, and adding whole-number exponents produces another whole number, so you can never accidentally create a variable in a denominator or under a radical. Integers behave the same way under those three operations, and just as integers are not closed under division, polynomials are not either:

\frac{x+1}{x}

is not a polynomial.

Key terms

Monomial.: A single term that is a number, a variable, or a product of numbers and variables with whole-number exponents, such as $-7x^2y$ .
Polynomial.: A sum of one or more monomials. Terms with negative or fractional exponents, or with a variable in a denominator, are not allowed.
Coefficient.: The numerical factor of a term, including its sign. In $-5x^3$ the coefficient is $-5$ .
Degree of a polynomial.: The greatest degree of any of its terms. For a one-variable term, the degree is the exponent on the variable; a constant has degree $0$ .
Standard form.: A polynomial written with terms in order from highest degree to lowest, as in $-2x^3 + x^2 + 4x - 9$ .
Like terms.: Terms with identical variable parts and identical exponents, so their coefficients can be added or subtracted. $3x^2$ and $-8x^2$ are like terms; $3x^2$ and $3x$ are not.
Leading coefficient.: The coefficient of the term with the highest degree once the polynomial is in standard form.
Closure.: The property that performing an operation on two members of a set always produces another member of that set. Polynomials are closed under addition, subtraction, and multiplication.

Worked example

Simplify completely and write the answer in standard form:

(3x - 4)(2x + 5) - (x^2 - 6x + 1)

. Then state the degree and the leading coefficient of the result.

Step 1 — Multiply the two binomials. Every term of

3x - 4

meets every term of

2x + 5

3x \cdot 2x = 6x^2

3x \cdot 5 = 15x

-4 \cdot 2x = -8x

-4 \cdot 5 = -20

.

So the product is

6x^2 + 15x - 8x - 20 = 6x^2 + 7x - 20

.

Step 2 — Rewrite the subtraction as adding the opposite. The minus sign must reach all three terms inside the second parentheses:

6x^2 + 7x - 20 - x^2 + 6x - 1

Notice

-(-6x)

became

+6x

and

-(+1)

became

-1

.

Step 3 — Combine like terms. The

x^2

terms:

6x^2 - x^2 = 5x^2

. The

x

terms:

7x + 6x = 13x

. The constants:

-20 - 1 = -21

5x^2 + 13x - 21

Step 4 — Answer the questions. The expression is already in standard form. Its degree is

2

and its leading coefficient is

5

.

Step 5 — Check with a value. Let

x = 2

. Original:

(6 - 4)(4 + 5) - (4 - 12 + 1) = (2)(9) - (-7) = 18 + 7 = 25

. Answer:

5(4) + 13(2) - 21 = 25

. The values match, so the simplification holds.

Practice questions

Which expression is equivalent to

(4x^2 - 3x + 8) - (x^2 + 5x - 2)

$3x^2 - 8x + 10$
$3x^2 + 2x + 6$
$5x^2 + 2x + 6$
$3x^2 - 8x + 6$

Answer: $3x^2 - 8x + 10$

Distribute the minus sign to all three terms of the second polynomial:

4x^2 - 3x + 8 - x^2 - 5x + 2

. Combining like terms gives

3x^2 - 8x + 10

. The choice

3x^2 + 2x + 6

comes from subtracting only the first term and then adding the rest, and

3x^2 - 8x + 6

comes from writing

8 - 2

instead of

8 + 2

for the constants. Checking with

x = 2

confirms the answer: the original gives

18 - 12 = 6

, and

12 - 16 + 10 = 6

Multiply and simplify:

-2x(3x^2 - x + 6) + (x + 4)(x - 4)

. Write your result in standard form and identify the degree.

Answer: $-6x^3 + 2x^2 - 12x + x^2 - 16 = -6x^3 + 3x^2 - 12x - 16$ ; degree 3.

First distribute the monomial:

-2x \cdot 3x^2 = -6x^3

-2x \cdot (-x) = +2x^2

, and

-2x \cdot 6 = -12x

, so the monomial part is

-6x^3 + 2x^2 - 12x

. Next multiply the binomials:

x \cdot x = x^2

x \cdot (-4) = -4x

4 \cdot x = 4x

4 \cdot (-4) = -16

, and the middle terms cancel to leave

x^2 - 16

. Adding the two results and combining the

x^2

terms gives

-6x^3 + 3x^2 - 12x - 16

. Work each product separately before combining, and recheck every sign — the term

-2x \cdot (-x)

is positive because a negative times a negative is positive. The highest power is

x^3

, so the degree is 3.

Ana says that because

x^2

and

x

both contain the variable

x

, the expression

5x^2 + 3x

simplifies to

8x^3

. Explain what is wrong with her reasoning and what the expression actually simplifies to.

Answer: Nothing simplifies; $5x^2 + 3x$ is already in simplest form. Ana confused adding like terms with multiplying terms.

Like terms must have the same variable raised to the same exponent. Since

x^2

and

x

have different exponents, their coefficients cannot be added. Ana also imported a rule from multiplication: exponents add only when you multiply, as in

5x^2 \cdot 3x = 15x^3

. Testing

x = 2

exposes the error quickly:

5(4) + 3(2) = 26

, but

8(8) = 64

. The expression

5x^2 + 3x

is a binomial of degree 2 and is already fully simplified.

FAQ

Do I have to use FOIL to multiply binomials?: No. FOIL is just a memory aid for the distributive property applied to two binomials, and it stops working the moment a factor has three or more terms. The box (area) method or simply distributing each term of the first factor across the whole second factor works for any sizes and makes it easier to see that you have used every pair of terms.
How do I know if an expression is not a polynomial?: Look at the exponents on the variables. Every exponent must be a whole number: $0, 1, 2, 3, \dots$ . So $x^{-2}$ , $\frac{7}{x}$ , $\sqrt{x}$ , and $x^{1/3}$ all disqualify an expression. Numbers in denominators are fine, though — $\frac{x}{4}$ is the same as $\frac{1}{4}x$ , which is a legitimate polynomial term.
Why does my answer have fewer terms than the number of products I found?: Because some of the products turn out to be like terms and merge. Multiplying two binomials produces four products, but the outer and inner products usually share the same variable power, so they combine into a single middle term. That is why $(2x - 3)(x + 5)$ gives the three-term answer $2x^2 + 7x - 15$ .
What is the fastest way to catch a sign mistake?: Substitute a convenient number such as $x = 2$ into both the original expression and your simplified answer. If the two values differ, you have an error, and it is almost always a minus sign that failed to reach every term inside a set of parentheses. Avoid using $x = 0$ or $x = 1$ alone, since those can hide certain mistakes.

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