Polynomial Addition and Subtraction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polynomial Addition and Subtraction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Adding or subtracting polynomials by combining like terms—terms with the same variable raised to the same power.

Think of like terms as the same type of object: $3x^2$ and $5x^2$ are both ' $x^2$ things,' so you can combine them into $8x^2$ , just like 3 apples plus 5 apples equals 8 apples. You cannot combine $x^2$ and $x$ any more than you can add apples and oranges.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Adding polynomials means combining only like terms — matching variable and exponent.

Common stuck point: The procedure for polynomial addition and subtraction is the easy part; the trap is adding exponents instead of coefficients. Asking "Am I joining two polynomials with $+$ or $-$ and merging only same-power terms?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I joining two polynomials with $+$ or $-$ and merging only same-power terms?

Worked Examples

Example 1

easy

Add

(3x^2 + 5x - 2) + (x^2 - 3x + 7)

Answer

4x^2 + 2x + 5

First step

Step 1: Group like terms:

(3x^2 + x^2) + (5x - 3x) + (-2 + 7)

Full solution

2
Step 2: Combine: $4x^2 + 2x + 5$ .
3
Check: At $x = 1$ : $(3+5-2) + (1-3+7) = 6 + 5 = 11$ and $4+2+5 = 11$ ✓

To add polynomials, combine like terms — terms with the same variable raised to the same power. The coefficients are added while the variable parts stay the same.

Example 2

medium

Subtract

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

Example 3

medium

Add

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

Example 4

medium

Find the perimeter of a rectangle whose length is

(3x + 5)

and width is

(2x - 1)

Example 5

medium

P(x) = 3x^2 - x + 4

and

Q(x) = -x^2 + 5x - 1

, find

P(x) + Q(x)

Example 6

medium

Simplify

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

Example 7

hard

Simplify

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

Example 8

hard

Add

(2x^{n+1} + 3x^n) + (x^{n+1} - 5x^n + x)

for integer

n \geq 1

Example 9

hard

Simplify

-[2x^2 - (3x + 1)] + (x^2 - 2x)

Example 10

hard

For what value of

k

(2x^2 + kx + 3) + (3x^2 - 4x + 1)

equal to

5x^2 + 4

Example 11

challenge

Let

f(x) = x^3 + 2x - 1

and

g(x) = -x^3 + x^2 + 5

. Compute

f(x) + g(x)

and evaluate it at

x = 2

Practice Problems

x+2

2x-1

, and

3x+4

. Find the perimeter polynomial.

Example 15

medium

Subtract

(5x^2y-3xy)-(2x^2y+xy)

Example 16

medium

Add

(\frac{1}{2}x^2+x)+(\frac{1}{2}x^2+2x)

Example 17

medium

Find the missing polynomial

P

P+(2x^2-x)=5x^2+3x

Example 18

medium

Simplify

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

Example 19

medium

Add

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

Example 20

challenge

Find a polynomial that, when added to

x^2-3x+2

, gives the zero polynomial.

Example 21

challenge

f(x)=2x^2-x+3

and

g(x)=x^2+4x-1

, compute

2f(x)-g(x)

Example 22

challenge

For what value of

k

does

(kx^2+3x)-(2x^2+x)

have no

x^2

(2x^2y + 3xy^2) + (5x^2y - xy^2)

Example 31

medium

A triangle has sides

(x + 2)

(2x - 1)

, and

(3x + 4)

. Find its perimeter.

Example 32

medium

Find

A - B

A = 4x^3 - 2x + 7

and

B = 4x^3 - 2x - 1

Example 33

hard

P(x) + Q(x) = 5x^2 - 3x + 2

and

P(x) = 3x^2 + x - 4

, find

Q(x)

Example 34

hard

If the sum of

3x^2 + ax + 7

and

bx^2 - 4x + c

equals

5x^2 - x + 10

, find

a

b

, and

c

Example 35

hard

Find the polynomial that must be added to

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

to get

5x^3 + 2x^2 - 7

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Related Concepts

Expressions Polynomials Polynomial Multiplication Factoring by Grouping

Background Knowledge

These ideas may be useful before you work through the harder examples.

expressionspolynomials