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# Multiplying Polynomials

## Distribute monomials, binomials, and trinomials

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Multiplying Polynomials

The length of a rectangular garden plot is x3+5x21\begin{align*}x^3 + 5x^2 - 1\end{align*}. The width of the plot is x2+3\begin{align*}x^2 + 3\end{align*}. What is the area of the garden plot?

### Multiplying Polynomials

Multiplying together polynomials is very similar to multiplying together factors. You can FOIL or we will also present an alternative method. When multiplying together polynomials, you will need to use the properties of exponents, primarily the Product Property (aman=am+n)\begin{align*}(a^m \cdot a^n = a^{m+n})\end{align*} and combine like terms.

Let's find the following products.

1. (x25)(x3+2x9)\begin{align*}(x^2-5)(x^3 + 2x-9)\end{align*}

Using the FOIL method, you need be careful. First, take the x2\begin{align*}x^2\end{align*} in the first polynomial and multiply it by every term in the second polynomial.

Now, multiply the -5 and multiply it by every term in the second polynomial.

Lastly, combine any like terms. In this example, only the x3\begin{align*}x^3\end{align*} terms can be combined.

1. (x2+4x7)(x38x2+6x11)\begin{align*}(x^2+4x-7)(x^3-8x^2+6x-11)\end{align*}

In this problem, we will use the “box” method. Align the two polynomials along the top and left side of a rectangle and make a row or column for each term. Write the polynomial with more terms along the top of the rectangle.

Multiply each term together and fill in the corresponding spot.

Finally, combine like terms. The final answer is x54x433x3+69x286x+77\begin{align*}x^5 -4x^4 -33x^3 + 69x^2 -86x + 77\end{align*}. This method presents an alternative way to organize the terms. Use whichever method you are more comfortable with. Keep in mind, no matter which method you use, you will multiply every term in the first polynomial by every term in the second.

1. (x5)(2x+3)(x2+4)\begin{align*}(x-5)(2x + 3)(x^2 + 4)\end{align*}

In this problem we have three binomials. When multiplying three polynomials, start by multiplying the first two binomials together.

(x5)(2x+3)=2x2+3x10x15=2x27x15\begin{align*}(x-5)(2x+3) &= 2x^2+3x-10x-15\\ &= {\color{red}2x^2-7x-15}\end{align*}

Now, multiply the answer by the last binomial.

(2x27x15)(x2+4)=2x4+8x27x328x15x260=2x47x37x228x60\begin{align*}({\color{red}2x^2-7x-15})(x^2+4) &= 2x^4+8x^2-7x^3-28x-15x^2-60\\ &= 2x^4-7x^3-7x^2-28x-60\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the area of the garden plot.

Recall that the area of a rectangle is A=lw\begin{align*}A = lw\end{align*}, where l is the length and w is the width. Therefore, we need to multiply.

A=(x3+5x21)(x2+3)=x5+3x3+5x4+15x2x23\begin{align*} A =(x^3 + 5x^2 - 1)(x^2 + 3)\\ = x^5 + 3x^3 + 5x^4 + 15x^2 - x^2 - 3\end{align*}

Now combine like terms and simplify. Be sure to write your answer in standard form

x5+3x3+5x4+(15x2x2)3=x5+3x3+5x4+14x23=x5+5x4+3x3+14x23\begin{align*}x^5 + 3x^3 + 5x^4 + (15x^2 - x^2) - 3\\ = x^5 + 3x^3 + 5x^4 + 14x^2 - 3\\ = x^5 + 5x^4 + 3x^3 + 14x^2 - 3\end{align*}

Therefore, the area of the garden plot is x5+5x4+3x3+14x23\begin{align*}x^5 + 5x^4 + 3x^3 + 14x^2 - 3\end{align*}.

#### Example 2

Find the product: 2x2(3x34x2+12x9)\begin{align*}-2x^2(3x^3-4x^2+12x-9)\end{align*}.

Use the distributive property to multiply 2x2\begin{align*}-2x^2\end{align*} by the polynomial.

2x2(3x34x2+12x9)=6x5+8x424x3+18x2\begin{align*}-2x^2(3x^3-4x^2+12x-9) = -6x^5+8x^4-24x^3+18x^2\end{align*}

#### Example 3

Find the product: (4x26x+11)(3x3+x2+8x10)\begin{align*}(4x^2-6x+11)(-3x^3+x^2+8x-10)\end{align*}.

Multiply each term in the first polynomial by each one in the second polynomial.

(4x26x+11)(3x3+x2+8x10)=12x5+4x4+32x340x2 +18x46x348x2+60x 33x3+11x2+88x110=12x5+22x47x377x2+148x110\begin{align*}(4x^2-6x+11)(-3x^3+x^2+8x-10) &= -12x^5+4x^4+32x^3-40x^2\\ & \qquad \qquad \ +18x^4-6x^3-48x^2+60x\\ & \qquad \qquad \qquad \quad \ -33x^3+11x^2+88x-110\\ &= -12x^5+22x^4-7x^3-77x^2+148x-110\end{align*}

#### Example 4

Find the product: (x21)(3x4)(3x+4)\begin{align*}(x^2-1)(3x-4)(3x+4)\end{align*}.

Multiply the first two binomials together.

(x21)(3x4)=3x34x23x+4\begin{align*}(x^2-1)(3x-4) = 3x^3-4x^2-3x+4\end{align*}

Multiply this product by the last binomial.

\begin{align*}(3x^3-4x^2-3x+4)(3x+4) &= 9x^4+12x^3-12x^3-16x^2-9x^2-12x+12x-16\\ &= 9x^4-25x^2-16\end{align*}

#### Example 5

Find the product: \begin{align*}(2x-7)^2\end{align*}.

The square indicates that there are two binomials. Expand this and multiply.

\begin{align*}(2x-7)^2 &= (2x-7)(2x-7)\\ &= 4x^2-14x-14x+49\\ &= 4x^2-28x+49\end{align*}

### Review

Find the product.

1. \begin{align*}5x(x^2-6x+8)\end{align*}
2. \begin{align*}-x^2(8x^3-11x+20)\end{align*}
3. \begin{align*}7x^3(3x^3-x^2+16x+10)\end{align*}
4. \begin{align*}(x^2+4)(x-5)\end{align*}
5. \begin{align*}(3x^2-4)(2x-7)\end{align*}
6. \begin{align*}(9-x^2)(x+2)\end{align*}
7. \begin{align*}(x^2+1)(x^2-2x-1)\end{align*}
8. \begin{align*}(5x-1)(x^3+8x-12)\end{align*}
9. \begin{align*}(x^2-6x-7)(3x^2-7x+15)\end{align*}
10. \begin{align*}(x-1)(2x-5)(x+8)\end{align*}
11. \begin{align*}(2x^2+5)(x^2-2)(x+4)\end{align*}
12. \begin{align*}(5x-12)^2\end{align*}
13. \begin{align*}-x^4(2x+11)(3x^2-1)\end{align*}
14. \begin{align*}(4x+9)^2\end{align*}
15. \begin{align*}(4x^3-x^2-3)(2x^2-x+6)\end{align*}
16. \begin{align*}(2x^3-6x^2+x+7)(5x^2+2x-4)\end{align*}
17. \begin{align*}(x^3+x^2-4x+15)(x^2-5x-6)\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.5.

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distributive property The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.