In these lessons, we will look at how to add and subtract polynomials.

**Related Pages**

Algebra Terms

Algebraic Expressions

Adding & Subtracting Functions

More Algebra Lessons

Grade 7 Math Lessons

The following diagram shows examples of adding and subtracting polynomials. Scroll down the page for more examples and solutions on how to add and subtract polynomials.

Adding polynomials involves combining like terms.

**Example:**

Add the polynomials 5x – 2 + y and –3y + 5x + 2

**Solution:**

5x – 2 + y + (–3y + 5x + 2)

= 5x + 5x + y – 3y – 2 + 2

= 10x – 2y

**Example:**

Find the sum of –7x^{3}y + 4x^{2}y^{2} – 2 and
4x^{3}y + 1 – 8x^{2}y^{2}

**Solution:**

–7x^{3}y + 4x^{2}y^{2}– 2 + 4x^{3}y + 1 – 8x^{2}y^{2}

= –7x^{3}y + 4x^{3}y + 4x^{2}y^{2}– 8x^{2}y^{2} – 2 + 1

= –3x^{3}y – 4x^{2}y^{2} – 1

To subtract polynomials, remember to distribute the – sign into all the terms in the parenthesis.

**Example:**

Simplify –4x + 7 – (5x – 3)

**Solution:**

–4x + 7 – (5x – 3)

= –4x + 7 – 5x + 3

= –9x + 10

**Example:**

Simplify (5x^{2} + 2) – (– 4x^{2} + 7) + (–
3x^{2} – 5)

**Solution:**

(5x^{2} + 2) – (– 4x^{2} + 7) + (– 3x^{2}– 5)

= 5x^{2} + 2 + 4x^{2}– 7 – 3x^{2}– 5

= 5x^{2} + 4x^{2}– 3x^{2} + 2 – 7 – 5

= 6x^{2} – 10

To add polynomials

- Combine like terms.
- Write in descending order.

**Examples:**

- (4x
^{2}+ 8x - 7) + (2x^{2}- 5x - 12) - (5 + 24y
^{3}- 7y^{2}) + (-6y^{3}+ 7y^{2}+ 5) - (t
^{2}- t + 5) + (7t^{2}- 4t - 20)

To subtract polynomials

- Rewrite the subtraction as addition.

To change subtraction to addition, we must add the opposite, or additive inverse. 2. Combine like terms. 3. Write in descending order.

**Examples:**

- (4x
^{2}+ 8x - 7) - (2x^{2}- 5x - 12) - (6x
^{4}+ 3x^{3}- 1) - (4x^{3}- 5x + 9) - (1.5y
^{3}+ 4.8y^{2}+ 12) - (y^{3}- 1.7y^{2}+ 2y)

**Examples:**

- (2x
^{5}- 6x^{3}- 12x^{2}- 4) + (-11x^{5}+ 8x + 2x^{2}+ 6) - (-9y
^{3}- 6y^{2}- 11x + 2) - (-9y^{4}- 8y^{3}+ 4x^{2}+ 2x)

Adding polynomials and subtracting polynomials is essentially combining like terms of polynomial expressions. When adding and subtracting polynomials, they can either be arranged vertically or grouped according to degree. A knowledge of polynomial vocabulary is important before adding and subtracting polynomials. Multiplying monomials and binomials is another type of operation with polynomials.

**Examples:**

- (4x
^{2}- 3x + 2) + (5x^{2}+ 2x - 7) - (5x
^{3}+ 7x^{2}- x) + (8x^{3}+ 4x - 5) - (8x
^{2}+ 2x) - (10x^{2}+ 2x - 9) - -(6x
^{3}- 4x) - (2x^{3}+ x^{2}-2x)

**Examples:**

- (x
^{2}+ 4x + 5) + (6x + 3) - 2(x
^{4}+ 5x) - 6(x^{4}+ 8x - 3)

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