Suppose a factory needs to increase the number of units it outputs. Currently it has \begin{align*}w\end{align*} workers, and on average, each worker outputs \begin{align*}u\end{align*} units. If it increases the number of workers by 100 and makes changes to its processes so that each worker outputs 20 more units on average, how many total units will it output? What would you have to do to find the answer?

### Multiplying Polynomials with Binomials

A binomial is a polynomial with two terms. The Distributive Property also applies for multiplying binomials. Let’s think of the first parentheses as one term. The Distributive Property says that the term in front of the parentheses multiplies with each term inside the parentheses separately. Then, we add the results of the products.

\begin{align*}(a+b)(c+d)=(a+b)\cdot c+(a+b)\cdot d\end{align*} Let’s rewrite this answer as \begin{align*}c\cdot (a+b)+d\cdot (a+b)\end{align*}.

We see that we can apply the Distributive Property on each of the parentheses in turn.

\begin{align*}c \cdot (a+b)+d\cdot (a+b)=c\cdot a+c \cdot b+d \cdot a+d \cdot b \ (\text{or} \ ca+cb+da+db)\end{align*}

What you should notice is that when multiplying any two polynomials, **every term in one polynomial is multiplied by every term in the other polynomial**.

#### Let's multiply and simplify the following expressions:

- \begin{align*}(2x+1)(x+3)\end{align*}

We must multiply each term in the first polynomial with each term in the second polynomial. First, multiply the first term in the first parentheses by all the terms in the second parentheses.

Now we multiply the second term in the first parentheses by all terms in the second parentheses and add them to the previous terms.

Now we can simplify.

\begin{align*}(2x)(x)+(2x)(3)+(1)(x)+(1)(3) & = 2x^2+6x+x+3\\ & = 2x^2+7x+3\end{align*}

- \begin{align*}(4x-5)(x^2+x-20)\end{align*}

Multiply the first term in the binomial by each term in the polynomial, and then multiply the second term in the monomial by each term in the polynomial: \begin{align*}& \quad (4x)(x^2)+(4x)(x)+(4x)(\text{-}20)+(\text{-}5)(x^2)+(\text{-}5)(x)+(\text{-}5)(\text{-}20) \\ &=4x^3+4x^2-80x-5x^2-5x+100\\ & = 4x^3-x^2-85x+100\end{align*}

#### Real-World Problems with Multiplication of Polynomials

We can use multiplication to find the area and volume of geometric shapes.

#### Let's find the area of the following figure:

We use the formula for the area of a rectangle: \begin{align*}\text{Area}=\text{length}\cdot\text{width}\end{align*}. For the big rectangle:

\begin{align*}\text{Length} & = B+3, \ \text{Width}=B+2\\ \text{Area} &= (B+3)(B+2)\\ & = B^2+2B+3B+6\\ & = B^2+5B+6\end{align*}

### Examples

#### Example 1

Earlier, you were told that a factory has \begin{align*}w\end{align*} workers and on average, every worker outputs \begin{align*}u\end{align*} units. If it increases the number of workers by 100 and makes changes to its processes so that each worker outputs 20 more units on average, how many total units will it output?

To find the total number of units the factory will output, we need to multiply the number of workers by the number of units each worker outputs. If the number of workers is increased by 100, then there are \begin{align*}w+100\end{align*} workers. If each worker outputs 20 more units on average, then the number of units each worker outputs is \begin{align*}u + 20\end{align*}. Multiply these two expressions together and simplify to find the total units the factory will output:

\begin{align*}output &=(w+100)(u+200)\\ &=wu + 200w + 100u + 20000\end{align*}

This expression cannot be simplified any further and the factory will output \begin{align*}wu + 200w +100u + 20000\end{align*} units.

#### Example 2

Find the volume of the following figure.

\begin{align*}The \ volume \ of \ this \ shape & = (area \ of \ the \ base) \cdot (height).\\ \text{Area of the base} & = x(x+2)\\ & = x^2+2x\end{align*}

\begin{align*}Volume&=(area \ of \ base ) \times height\\ Volume&=(x^2+2x)(2x+1)\end{align*}

Now, multiply the two binomials together.

\begin{align*}Volume&=(x^2+2x)(2x+1)\\ &= x^2\cdot 2x+x^2\cdot 1+2x\cdot 2x+ 2x\cdot 1\\ &= 2x^3+x^2+2x^2+2x\\ &=2x^3+3x^2+2x\end{align*}

### Review

Multiply and simplify.

- \begin{align*}(x-2)(x+3)\end{align*}
- \begin{align*}(a+2)(2a)(a-3)\end{align*}
- \begin{align*}(-4xy)(2x^4 yz^3 -y^4 z^9)\end{align*}
- \begin{align*}(x-3)(x+2)\end{align*}
- \begin{align*}(a^2+2)(3a^2-4)\end{align*}
- \begin{align*}(7x-2)(9x-5)\end{align*}
- \begin{align*}(2x-1)(2x^2-x+3)\end{align*}
- \begin{align*}(3x+2)(9x^2-6x+4)\end{align*}
- \begin{align*}(a^2+2a-3)(a^2-3a+4)\end{align*}
- \begin{align*}(3m+1)(m-4)(m+5)\end{align*}

Find the areas of the following figures.

Find the volumes of the following figures.

**Mixed Review**

- Simplify \begin{align*}5x(3x+5)+11(-7-x)\end{align*}.
- Cal High School has grades nine through twelve. Of the school's student population, \begin{align*}\frac{1}{4}\end{align*} are freshmen, \begin{align*}\frac{2}{5}\end{align*} are sophomores, \begin{align*}\frac{1}{6}\end{align*} are juniors, and 130 are seniors. To the nearest whole person, how many students are in the sophomore class?
- Kerrie is working at a toy store and must organize 12 bears on a shelf. In how many ways can this be done?
- Find the slope between \begin{align*}\left ( \frac{3}{4},1 \right )\end{align*} and \begin{align*}\left ( \frac{3}{4}, -16 \right )\end{align*}.
- If \begin{align*}1 \ lb=454 \ grams\end{align*}, how many kilograms does a 260-pound person weigh?
- Solve for \begin{align*}v\end{align*}: \begin{align*}|16-v|=3\end{align*}.
- Is \begin{align*}y=x^4+3x^2+2\end{align*} a function? Use the definition of a function to explain.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.4.