Adding, subtracting and multiplying polynomials (FOIL)

The key idea

Words you will see
Word	Meaning	In $3x^2 - 5x + 2$
term	one piece between plus and minus signs	$3x^2$ , $-5x$ , $2$
coefficient	the number in front of a variable	3 and $-5$
like terms	same variable with the same exponent	$3x^2$ and $7x^2$ are like; $3x^2$ and $3x$ are not
degree	the highest exponent	2
standard form	terms written from highest exponent to lowest	already in standard form

Multiplying uses the distributive property over and over. For two binomials, there are four products:

(a + b)(c + d) = ac + ad + bc + bd

Three products come up so often that they are worth memorizing:

(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)(a - b) = a^2 - b^2

Worked examples

Example 1: subtracting polynomials

Problem Simplify $(4x^2 - 3x + 7) - (x^2 + 5x - 2)$ .

The minus sign in front of the second group changes the sign of every term inside it.
$(4x^2 - 3x + 7) - (x^2 + 5x - 2) = 4x^2 - 3x + 7 - x^2 - 5x + 2$
Group like terms, then combine them.
$4x^2 - x^2 - 3x - 5x + 7 + 2 = 3x^2 - 8x + 9$

Answer $3x^2 - 8x + 9$

Example 2: FOIL

Problem Multiply $(2x + 3)(x - 4)$ .

First, Outer, Inner, Last: $2x \cdot x$ , $2x \cdot (-4)$ , $3 \cdot x$ , $3 \cdot (-4)$ .
$(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12$
Combine the two middle terms.
$2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$

Answer $2x^2 - 5x - 12$

Example 3: a binomial times a trinomial

Problem Multiply $(x - 2)(x^2 + 3x - 1)$ .

FOIL only covers two binomials. Here, multiply each of the 2 terms by each of the 3 terms, so expect 6 products. First the $x$ , then the $-2$ .
$(x - 2)(x^2 + 3x - 1) = x^3 + 3x^2 - x - 2x^2 - 6x + 2$
Combine like terms and write in standard form.
$x^3 + 3x^2 - x - 2x^2 - 6x + 2 = x^3 + x^2 - 7x + 2$

Answer $x^3 + x^2 - 7x + 2$

Example 4 (test-hard): special products together

Problem Simplify $(3x - 5)^2 - (3x + 5)(3x - 5)$ .

Square of a difference: $a = 3x$ and $b = 5$ , so $a^2 - 2ab + b^2$ .
$(3x - 5)^2 = 9x^2 - 30x + 25$
Sum times difference: $a^2 - b^2$ .
$(3x + 5)(3x - 5) = 9x^2 - 25$
Subtract the second result, keeping it in parentheses so both of its signs change.
$9x^2 - 30x + 25 - (9x^2 - 25) = -30x + 50$

Answer $-30x + 50$

Common mistakes

Squaring a binomial without the middle term. $(x + 5)^2$ is $x^2 + 10x + 25$ , not $x^2 + 25$ . Fix: write it as $(x + 5)(x + 5)$ and FOIL.
Changing only the first sign when subtracting. $-(x^2 + 5x - 2)$ is $-x^2 - 5x + 2$ . All three signs change.
Combining unlike terms. $3x^2 + 2x$ stays as it is. It is not $5x^3$ or $5x^2$ .
Multiplying exponents when you multiply terms. $x^2 \cdot x^3 = x^5$ . Exponents add when you multiply.
Missing a product. A 2-term times a 3-term polynomial has $2 \times 3 = 6$ products before combining. Count them.

Quick methods

Practice

5 practice questions

Simplify $(5x^2 + 2x - 1) + (3x^2 - 6x + 4)$ .
1. $8x^2 - 4x + 3$
2. $8x^4 - 4x^2 + 3$
3. $8x^2 + 8x + 3$
4. $2x^2 - 4x + 3$
Show answer
Answer: $8x^2 - 4x + 3$
Add like terms: $5x^2 + 3x^2 = 8x^2$ , $2x - 6x = -4x$ , $-1 + 4 = 3$ . Adding never changes exponents, so $8x^4$ is wrong.
Simplify $(3x^2 - x + 2) - (x^2 - 4x + 7)$ .
1. $2x^2 + 3x - 5$
2. $2x^2 - 5x - 5$
3. $2x^2 - 5x + 9$
4. $4x^2 - 5x + 9$
Show answer
Answer: $2x^2 + 3x - 5$
Change every sign in the second group: $3x^2 - x + 2 - x^2 + 4x - 7$ . Combine: $2x^2 + 3x - 5$ . The choices with $-5x$ did not flip the sign of $-4x$ , and $+9$ did not flip the sign of 7.
Multiply $(x + 6)(x - 2)$ .
1. $x^2 - 12$
2. $x^2 + 4x - 12$
3. $x^2 - 4x - 12$
4. $x^2 + 8x - 12$
Show answer
Answer: $x^2 + 4x - 12$
FOIL: $x^2 - 2x + 6x - 12 = x^2 + 4x - 12$ . The choice $x^2 - 12$ multiplies only first and last terms and skips the middle.
Expand $(2x + 7)^2$ .
1. $4x^2 + 49$
2. $2x^2 + 28x + 49$
3. $4x^2 + 14x + 49$
4. $4x^2 + 28x + 49$
Show answer
Answer: $4x^2 + 28x + 49$
Use $a^2 + 2ab + b^2$ with $a = 2x$ and $b = 7$ : $4x^2 + 2(2x)(7) + 49 = 4x^2 + 28x + 49$ . The choice with $14x$ forgot the 2 in $2ab$ , and $4x^2 + 49$ has no middle term at all.
Multiply $(4x - 3)(4x + 3)$ .
1. $16x^2 - 9$
2. $16x^2 + 9$
3. $16x^2 - 24x - 9$
4. $8x^2 - 9$
Show answer
Answer: $16x^2 - 9$
This is $(a - b)(a + b) = a^2 - b^2$ . The middle terms $12x$ and $-12x$ cancel, leaving $16x^2 - 9$ . The choice $8x^2$ doubles $4x$ instead of squaring it.

Frequently asked questions

What does FOIL mean?

FOIL stands for First, Outer, Inner, Last: the four products you get when multiplying two binomials like $(2x + 3)(x - 4)$ . It is just the distributive property in a fixed order. For anything bigger than two binomials, use distribution or the box method instead.

What are like terms?

Like terms have exactly the same variables raised to exactly the same powers. $4x^2$ and $-x^2$ are like terms, so they combine to $3x^2$ . $4x^2$ and $4x$ are not, so they stay separate. Only the coefficients change when you combine.

What is the degree of a polynomial?

It is the highest exponent on the variable after you simplify. $x^3 + x^2 - 7x + 2$ has degree 3. The degree tells you the general shape of the graph and, later, the most solutions an equation can have.

How do I multiply polynomials with more than two terms?

Multiply every term in the first polynomial by every term in the second. A 2-term times a 3-term polynomial gives 6 products, and a 3-term times a 3-term gives 9. Then combine like terms. A box grid keeps track so you do not miss one.