Suppose that two cars are having a race. The distance traveled by one car after \begin{align*}t\end{align*} seconds is \begin{align*}10t^2 + 50t\end{align*} meters, while the distance traveled by the other car after \begin{align*}t\end{align*} seconds is \begin{align*}15t^2 + 40t\end{align*} meters. How far apart would the two cars be after \begin{align*}t\end{align*} seconds? What would you have to do to find the answer to this question?

### Adding and Subtracting Polynomials

To add or subtract polynomials, you have to group the like terms together and combine them to simplify.

#### Let's complete the following problems:

- Add \begin{align*}3x^2-4x+7\end{align*} and \begin{align*}2x^3-4x^2-6x+5\end{align*}.

\begin{align*}(3x^2-4x+7)+(2x^3-4x^2-6x+5)&=2x^3+(3x^2-4x^2 )+(-4x-6x)+(7+5)\\ &=2x^3-x^2-10x+12\end{align*}

- Subtract \begin{align*}5b^2-2a^2\end{align*} from \begin{align*}4a^2-8ab-9b^2\end{align*}.

\begin{align*}(4a^2-8ab-9b^2)-(5b^2-2a^2)&=[(4a^2- (-2a^2)]+(-9b^2-5b^2)-8ab\\ & = 6a^2-14b^2-8ab\end{align*}

#### Solving Real-World Problems Using Addition or Subtraction of Polynomials

Polynomials are useful for finding the areas of geometric objects. In the following problems, you will see this usefulness in action.

#### Let's write a polynomial that represents the area of each figure shown.

The blue square has the following area: \begin{align*}y \cdot y=y^2\end{align*}.

The yellow square has the following area: \begin{align*}x \cdot x = x^2\end{align*}.

The pink rectangles each have the following area: \begin{align*}x \cdot y =xy\end{align*}.

\begin{align*}\text{Test area} & = y^2+x^2+xy+xy\\ & = y^2 + x^2 + 2xy\end{align*}

To find the area of the green region we find the area of the big square and subtract the area of the little square.

The big square has an area of \begin{align*}y \cdot y =y^2\end{align*}.

The little square has an area of \begin{align*}x \cdot x = x^2\end{align*}.

Area of the green region \begin{align*}= y^2-x^2\end{align*}

### Examples

#### Example 1

Earlier, you were told that in a race, the distanced traveled by one car after \begin{align*}t\end{align*} seconds is \begin{align*}10t^2 + 50t\end{align*} meters while the distance traveled by the second car after \begin{align*}t\end{align*} seconds is \begin{align*}15t^2 + 40t\end{align*} meters. How far apart would the two cars be after \begin{align*}t\end{align*} seconds.

To solve this question, you need to subtract the two expressions that represent distances. Let's subtract the distance of the first car from the distance of the second car and simplify.

\begin{align*}15t^2 + 40t - (10t^2 + 50t) &=15t^2 + 40t - 10t^2 - 50t & \text{Distribute the minus sign}\\ &=5t^2 -10t &\text{Combine like terms}\end{align*}

The two cars would be \begin{align*}5t^2-10t\end{align*} meters apart after \begin{align*}t\end{align*} seconds.

#### Example 2

Subtract \begin{align*} 4t^2+7t^3-3t-5\end{align*} from \begin{align*}6t+3-5t^3+9t^2\end{align*}.

When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:

\begin{align*}6t+3-5t^3+9t^2-(4t^2+7t^3-3t-5)&= \\ 6t+3-5t^3+9t^2-(4t^2)-(7t^3)-(-3t)-(-5)&=\\ 6t+3-5t^3+9t^2-4t^2-7t^3+3t+5&=\\ (6t+3t)+(3+5)+(-5t^3-7t^3)+(9t^2-4t^2)&=\\ 9t+8-12t^3+5t^2&=\\ -12t^3+5t^2+9t+8\\ \end{align*}

The final answer is in standard form.

### Review

Add and simplify.

- \begin{align*}(x+8)+(-3x-5)\end{align*}
- \begin{align*}(8r^4-6r^2-3r+9)+(3r^3+5r^2+12r-9)\end{align*}
- \begin{align*}(-2x^2+4x-12) + (7x+x^2)\end{align*}
- \begin{align*}(2a^2b-2a+9)+(5a^2b-4b+5)\end{align*}
- \begin{align*}(6.9a^2-2.3b^2+2ab)+(3.1a-2.5b^2+b)\end{align*}

Subtract and simplify.

- \begin{align*}(-t+15t^2)-(5t^2+2t-9)\end{align*}
- \begin{align*}(-y^2+4y-5)-(5y^2+2y+7)\end{align*}
- \begin{align*}(-h^7+2h^5+13h^3+4h^2-h-1)-(-3h^5+20h^3-3h^2+8h-4)\end{align*}
- \begin{align*}(-5m^2-m)-(3m^2+4m-5)\end{align*}
- \begin{align*}(2a^2b-3ab^2+5a^2b^2)-(2a^2b^2+4a^2b-5b^2)\end{align*}

Find the area of the following figures.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.2.