# How do you multiply (4x^2-7x-1)(6x^2+x-7)?

Jan 9, 2017

$24 {x}^{4} - 38 {x}^{3} - 41 {x}^{2} + 48 x + 7$

#### Explanation:

Your starting expression looks like this

$\left(4 {x}^{2} - 7 x - 1\right) \cdot \left(6 {x}^{2} + x - 7\right)$

In order to multiply all the terms listed here, you must use the distributive property.

The trick to using the distributive property is to pick the terms listed in one of the two parentheses as reference terms. I'll pick the terms listed in the second parentheses as reference terms and assign $\textcolor{b l u e}{1}$, $\textcolor{b l u e}{2}$, and $\textcolor{b l u e}{3}$ to help keep track of the multiplications

$\stackrel{\textcolor{b l u e}{1}}{6 {x}^{2}} + \stackrel{\textcolor{b l u e}{2}}{x} - \stackrel{\textcolor{b l u e}{3}}{7} \text{ } \to$ reference terms

Now pick the terms listed in the first set of parentheses and multiply them by the reference terms.

(color(red)(4x^2) color(purple)(-7x) color(brown)(-1)) * [ $\stackrel{\textcolor{b l u e}{1}}{6 {x}^{2}} + \stackrel{\textcolor{b l u e}{2}}{x} + \stackrel{\textcolor{b l u e}{3}}{\left(- 7\right)}$ ]

$\textcolor{red}{4 {x}^{2}} \cdot \stackrel{\textcolor{b l u e}{1}}{6 {x}^{2}} = 24 {x}^{4}$

$\textcolor{red}{4 {x}^{2}} \cdot \stackrel{\textcolor{b l u e}{2}}{x} = 4 {x}^{3}$

$\textcolor{red}{4 {x}^{2}} \cdot \stackrel{\textcolor{b l u e}{3}}{\left(- 7\right)} = - 28 {x}^{2}$

then continue with the second one

$\textcolor{p u r p \le}{- 7 x} \cdot \stackrel{\textcolor{b l u e}{1}}{6 {x}^{2}} = - 42 {x}^{3}$

$\textcolor{p u r p \le}{- 7 x} \cdot \stackrel{\textcolor{b l u e}{2}}{x} = - 7 {x}^{2}$

$\textcolor{p u r p \le}{- 7 x} \cdot \stackrel{\textcolor{b l u e}{3}}{\left(- 7\right)} = 49 x$

and the third one

$\textcolor{b r o w n}{- 1} \cdot \stackrel{\textcolor{b l u e}{1}}{6 {x}^{2}} = - 6 {x}^{2}$

$\textcolor{b r o w n}{- 1} \cdot \stackrel{\textcolor{b l u e}{2}}{x} = - x$

$\textcolor{b r o w n}{- 1} \cdot \stackrel{\textcolor{b l u e}{3}}{\left(- 7\right)} = 7$

Now put the nine terms that you got together to get

$24 {x}^{4} + 4 {x}^{3} - 28 {x}^{2} - 42 {x}^{3} - 7 {x}^{2} + 49 x - 6 {x}^{2} - x + 7$

Group like terms

$24 {x}^{4} + \left(4 {x}^{3} - 42 {x}^{3}\right) + \left(- 28 {x}^{2} - 7 {x}^{2} - 6 {x}^{2}\right) + \left(49 x - x\right) + 7$

to get

$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{24 {x}^{4} - 38 {x}^{3} - 41 {x}^{2} + 48 x + 7}}}$

Jan 9, 2017

$\left(4 {x}^{2} - 7 x - 1\right) \left(6 {x}^{2} + x - 7\right)$

$= 24 {x}^{4} - 38 {x}^{3} - 41 {x}^{2} + 48 x + 7$.

#### Explanation:

Multiplying polynomials is similar to multiplying regular numbers— each piece from the first polynomial needs to get multiplied by each piece in the second, and then these smaller products get added together to form the final product.

You may remember multiplication of three-digit numbers like this:

$\textcolor{w h i t e}{\times} 123$
$\underline{\times 456}$

You'd multiply the 6 by the 3, 2, and 1; then move over a column and multiply the 5 by the 3, 2, and 1; and finally multiply the 4 with the 3, 2, and 1.

It's like you're pairing each digit of 123 with each digit of 456, creating 9 smaller multiplications. What this method really does is this:

$\left(100 + 20 + 3\right) \times \left(400 + 50 + 6\right)$

Each piece in the left bracket is paired with each piece in the right bracket, creating 9 smaller products; these are ultimately added together to give us a final product.

Multiplying polynomials is done the exact same way. We pair off each term in the left polynomial with each one in the right, creating smaller products; the sum of these smaller products will be our final product.

So, we have

$\left(4 {x}^{2} - 7 x - 1\right) \left(6 {x}^{2} + x - 7\right)$

$= \textcolor{w h i t e}{+} \left(4 {x}^{2} \cdot 6 {x}^{2}\right) + \left(4 {x}^{2} \cdot x\right) + \left(4 {x}^{2} \cdot \text{-} 7\right)$
$\textcolor{w h i t e}{=} + \left(\text{-"7x*6x^2)+("-"7x*x)" "+("-"7x*"-} 7\right)$
color(white)=+"  "("-"1*6x^2)+"  "("-"1*x)"  "+"  "("-"1*"-"7)

$= 24 {x}^{4} + 4 {x}^{3} - 28 {x}^{2}$
$\textcolor{w h i t e}{=} - 42 {x}^{3} - 7 {x}^{2} + 49 x$
$\textcolor{w h i t e}{=} - 6 {x}^{2} - x + 7$

Now, we just add the like terms, and we get

$= 24 {x}^{4} - 38 {x}^{3} - 41 {x}^{2} + 48 x + 7$