Lesson Objectives

- Learn how to Add Polynomials
- Learn how to Subtract Polynomials

## Adding Polynomials

To add polynomials, we simply combine like terms. Let's take a look at a few examples.

Example 1: Simplify each. $$(8x^2+9x-9) + (13x^2+4x- 1)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$8x^2 + 13x^2 + 9x + 4x - 9 - 1$$ Now, we can combine like terms: $$21x^2 + 13x - 10$$ Example 2: Simplify each. $$(12x^4 - 6x^2 + 12x) + (-11x^2 + 11x + 7) + (-4x^4 + 10x^2 - 12x)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$12x^4 - 4x^4 - 6x^2 - 11x^2 + 10x^2 + 12x + 11x - 12x + 7$$ Now, we can combine like terms: $$8x^4 - 7x^2 + 11x + 7$$ Example 3: Simplify each. $$(4x^4y^2 + 2x^2y + 5x^3 + 2x) + (-6x^3 + 5y^3 + 8x^2 + 7x) + (x^2 - 2x^2y)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$4x^4y^2 + 2x^2y - 2x^2y + 5x^3 - 6x^3 + 5y^3 + 8x^2 + x^2 + 2x + 7x$$ Now, we can combine like terms: $$4x^4y^2 -x^3 + 5y^3 + 9x^2 + 9x$$

Example 4: Simplify each. $$(2x^4 + 2x^2 - 4) - (x^4 + 7x^2 - 6) - (-3x^4 + 2x^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^4 + 2x^2 - 4) + (-1)(x^4 + 7x^2 - 6) + (-1)(-3x^4 + 2x^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^4 + 2x^2 - 4 - x^4 - 7x^2 + 6 + 3x^4 - 2x^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$2x^4 - x^4 + 3x^4 + 2x^2 - 7x^2 - 2x^2 - 4 + 6$$ Now, we can combine like terms: $$4x^4 - 7x^2 + 2$$ Example 5: Simplify each. $$(2x^3y^2 + 4 - 6y^4) - (3x^3y^2 - 3x^4y^4 - 4) - (2 - 7y^4 + 5x^3y^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^3y^2 + 4 - 6y^4) + (-1)(3x^3y^2 - 3x^4y^4 - 4) + (-1)(2 - 7y^4 + 5x^3y^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^3y^2 + 4 - 6y^4 - 3x^3y^2 + 3x^4y^4 + 4 - 2 + 7y^4 - 5x^3y^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$3x^4y^4 + 2x^3y^2 - 3x^3y^2 - 5x^3y^2 - 6y^4 + 7y^4 + 4 + 4 - 2$$ Now, we can combine like terms: $$3x^4y^4 - 6x^3y^2 + y^4 + 6$$

Example 1: Simplify each. $$(8x^2+9x-9) + (13x^2+4x- 1)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$8x^2 + 13x^2 + 9x + 4x - 9 - 1$$ Now, we can combine like terms: $$21x^2 + 13x - 10$$ Example 2: Simplify each. $$(12x^4 - 6x^2 + 12x) + (-11x^2 + 11x + 7) + (-4x^4 + 10x^2 - 12x)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$12x^4 - 4x^4 - 6x^2 - 11x^2 + 10x^2 + 12x + 11x - 12x + 7$$ Now, we can combine like terms: $$8x^4 - 7x^2 + 11x + 7$$ Example 3: Simplify each. $$(4x^4y^2 + 2x^2y + 5x^3 + 2x) + (-6x^3 + 5y^3 + 8x^2 + 7x) + (x^2 - 2x^2y)$$ Since we have addition, we can simply drop the parentheses. Let's rewrite the problem and arrange our like terms next to each other: $$4x^4y^2 + 2x^2y - 2x^2y + 5x^3 - 6x^3 + 5y^3 + 8x^2 + x^2 + 2x + 7x$$ Now, we can combine like terms: $$4x^4y^2 -x^3 + 5y^3 + 9x^2 + 9x$$

### Subtracting Polynomials

When we subtract one polynomial from another, we change the subtraction sign to addition and change the sign of each term of the polynomial that is being subtracted away. In some cases, we will see tutorials where the sign is changed from "-" to "+" and then a "-1" is written outside of the polynomial that is being subtracted away. This "-1" just serves as a reminder. When we distribute the "-1" to each term inside of the parentheses, we are changing the sign of each term.Example 4: Simplify each. $$(2x^4 + 2x^2 - 4) - (x^4 + 7x^2 - 6) - (-3x^4 + 2x^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^4 + 2x^2 - 4) + (-1)(x^4 + 7x^2 - 6) + (-1)(-3x^4 + 2x^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^4 + 2x^2 - 4 - x^4 - 7x^2 + 6 + 3x^4 - 2x^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$2x^4 - x^4 + 3x^4 + 2x^2 - 7x^2 - 2x^2 - 4 + 6$$ Now, we can combine like terms: $$4x^4 - 7x^2 + 2$$ Example 5: Simplify each. $$(2x^3y^2 + 4 - 6y^4) - (3x^3y^2 - 3x^4y^4 - 4) - (2 - 7y^4 + 5x^3y^2)$$ Let's begin by changing each subtraction into addition. We will also change each term that is being subtracted away into its opposite. To make this process clear, let's change the sign from "-" to "+" and place a "-1" outside of the parentheses: $$(2x^3y^2 + 4 - 6y^4) + (-1)(3x^3y^2 - 3x^4y^4 - 4) + (-1)(2 - 7y^4 + 5x^3y^2)$$ Now, let's distribute the -1 to each term inside of the parentheses. This will change the sign of each term that is being subtracted away: $$2x^3y^2 + 4 - 6y^4 - 3x^3y^2 + 3x^4y^4 + 4 - 2 + 7y^4 - 5x^3y^2$$ Now, we just have addition. We can rewrite the problem and arrange our like terms next to each other: $$3x^4y^4 + 2x^3y^2 - 3x^3y^2 - 5x^3y^2 - 6y^4 + 7y^4 + 4 + 4 - 2$$ Now, we can combine like terms: $$3x^4y^4 - 6x^3y^2 + y^4 + 6$$

#### Skills Check:

Example #1

Simplify each.

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

Please choose the best answer.

A

$$4x^4 + 4x^3$$

B

$$2x^4 + 4x^3$$

C

$$5x^4 + 4x^3$$

D

$$-x^4 - 2x^3$$

E

$$x^4 + 2x^3$$

Example #2

Simplify each.

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

Please choose the best answer.

A

$$x^3 - 3x - 8$$

B

$$x^3 - 3x - 7$$

C

$$2x^3 - 3x - 7$$

D

$$x^3 - 3x - 12$$

E

$$-2x^3 - 5x + 1$$

Example #3

Simplify each.

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

Please choose the best answer.

A

$$-4x^4 - 2x^2$$

B

$$-x^4 - 2x^2$$

C

$$-x^2$$

D

$$3x^2$$

E

$$5x^4 - x^2$$

Example #4

Simplify each.

$$(5x^3y^3 + x^3y + 5y^4) + (y^4 + 3x^3y^4 + 3x^3y)$$

Please choose the best answer.

A

$$-x^3y^4 + 3x^3y^3 + 4x^3y + 6y^4$$

B

$$-x^3y^4 + 3x^3y^3 + 4x^3y + 3y^4$$

C

$$3x^3y^4 + 5x^3y^3 + 4x^3y + 6y^4$$

D

$$3x^3y^4 + 3x^3y^3 + 4x^3y + 6y^4$$

E

$$-3x^3y^4 + 3x^3y^3 - 2x^3y + 6y^4$$

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