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# 6.5: Multiplying Polynomials

Difficulty Level: At Grade Created by: CK-12
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Practice Multiplying Polynomials
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Estimated21 minsto complete
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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?

### Guidance

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.

#### Example A

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

Solution: 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.

#### Example B

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

Solution: In this example, 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.

#### Example C

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

Solution: In this example 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*}

Intro Problem Revisit Recall that the area of a rectangle is A=lw\begin{align*}A = lw\end{align*}, where l is the lenght 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*}.

### Guided Practice

Find the product of the polynomials.

1. 2x2(3x34x2+12x9)\begin{align*}-2x^2(3x^3-4x^2+12x-9)\end{align*}

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

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

4. (2x7)2\begin{align*}(2x-7)^2\end{align*}

1. 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*}

2. 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*}

3. 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.

(3x34x23x+4)(3x+4)=9x4+12x312x316x29x212x+12x16=9x425x216\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*}

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

(2x7)2=(2x7)(2x7)=4x214x14x+49=4x228x+49\begin{align*}(2x-7)^2 &= (2x-7)(2x-7)\\ &= 4x^2-14x-14x+49\\ &= 4x^2-28x+49\end{align*}

### Practice

Find the product.

1. 5x(x26x+8)\begin{align*}5x(x^2-6x+8)\end{align*}
2. x2(8x311x+20)\begin{align*}-x^2(8x^3-11x+20)\end{align*}
3. 7x3(3x3x2+16x+10)\begin{align*}7x^3(3x^3-x^2+16x+10)\end{align*}
4. (x2+4)(x5)\begin{align*}(x^2+4)(x-5)\end{align*}
5. (3x24)(2x7)\begin{align*}(3x^2-4)(2x-7)\end{align*}
6. (9x2)(x+2)\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*}

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### Vocabulary Language: English

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, $a(b + c) = ab + ac$.

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