# Multiplication of polynomials  (Page 2/2)

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$\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$

This method is commonly called the FOIL method .

• F First terms
• O Outer terms
• I Inner terms
• L Last terms

$\left(a+b\right)\left(2+3\right)=\underset{2\text{\hspace{0.17em}}\text{terms}}{\underbrace{\left(a+b\right)+\left(a+b\right)}}+\underset{3\text{\hspace{0.17em}}\text{terms}}{\underbrace{\left(a+b\right)+\left(a+b\right)+\left(a+b\right)}}$

Rearranging,

$\begin{array}{l}=a+a+b+b+a+a+a+b+b+b\\ =2a+2b+3a+3b\end{array}$

Combining like terms,

$=5a+5b$

This use of the distributive property suggests the following rule.

## Multiplying a polynomial by a polynomial

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

## Sample set c

Perform the following multiplications and simplify. With some practice, the second and third terms can be combined mentally.   $\begin{array}{lll}{\left(m-3\right)}^{2}\hfill & =\hfill & \left(m-3\right)\left(m-3\right)\hfill \\ \hfill & =\hfill & m\cdot m+m\left(-3\right)-3\cdot m-3\left(-3\right)\hfill \\ \hfill & =\hfill & {m}^{2}-3m-3m+9\hfill \\ \hfill & =\hfill & {m}^{2}-6m+9\hfill \end{array}$

$\begin{array}{llll}{\left(x+5\right)}^{3}\hfill & =\hfill & \left(x+5\right)\left(x+5\right)\left(x+5\right)\hfill & \text{Associate}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{two}\text{\hspace{0.17em}}\text{factors}\text{.}\hfill \\ \hfill & =\hfill & \left[\left(x+5\right)\left(x+5\right)\right]\left(x+5\right)\hfill & \hfill \\ \hfill & =\hfill & \left[{x}^{2}+5x+5x+25\right]\left(x+5\right)\hfill & \hfill \\ \hfill & =\hfill & \left[{x}^{2}+10x+25\right]\left(x+5\right)\hfill & \hfill \\ \hfill & =\hfill & {x}^{2}\cdot x+{x}^{2}\cdot 5+10x\cdot x+10x\cdot 5+25\cdot x+25\cdot 5\hfill & \hfill \\ \hfill & =\hfill & {x}^{3}+5{x}^{2}+10{x}^{2}+50x+25x+125\hfill & \hfill \\ \hfill & =\hfill & {x}^{3}+15{x}^{2}+75x+125\hfill & \hfill \end{array}$

## Practice set c

Find the following products and simplify.

$\left(a+1\right)\left(a+4\right)$

${a}^{2}+5a+4$

$\left(m-9\right)\left(m-2\right)$

${m}^{2}-11m+18$

$\left(2x+4\right)\left(x+5\right)$

$2{x}^{2}+14x+20$

$\left(x+y\right)\left(2x-3y\right)$

$2{x}^{2}-xy-3{y}^{2}$

$\left(3{a}^{2}-1\right)\left(5{a}^{2}+a\right)$

$15{a}^{4}+3{a}^{3}-5{a}^{2}-a$

$\left(2{x}^{2}{y}^{3}+x{y}^{2}\right)\left(5{x}^{3}{y}^{2}+{x}^{2}y\right)$

$10{x}^{5}y{}^{5}+7{x}^{4}{y}^{4}+{x}^{3}{y}^{3}$

$\left(a+3\right)\left({a}^{2}+3a+6\right)$

${a}^{3}+6{a}^{2}+15a+18$

$\left(a+4\right)\left(a+4\right)$

${a}^{2}+8a+16$

$\left(r-7\right)\left(r-7\right)$

${r}^{2}-14r+49$

${\left(x+6\right)}^{2}$

${x}^{2}+12x+36$

${\left(y-8\right)}^{2}$

${y}^{2}-16y+64$

## Sample set d

Perform the following additions and subtractions.

$\begin{array}{ll}3x+7+\left(x-3\right).\hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{parentheses}\text{.}\text{\hspace{0.17em}}\text{They}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{preceded}\text{\hspace{0.17em}}\text{by}\hfill \\ \hfill & \text{a}\text{\hspace{0.17em}}"+"\text{\hspace{0.17em}}\text{sign,}\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{them}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{leave}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{​}\text{each}\hfill \\ \hfill & \text{term}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ 3x+7+x-3\hfill & \text{Combine}\text{\hspace{0.17em}}\text{like}\text{\hspace{0.17em}}\text{terms}\text{.}\hfill \\ 4x+4\hfill & \hfill \end{array}\text{\hspace{0.17em}}$

$\begin{array}{ll}5{y}^{3}+11-\left(12{y}^{3}-2\right).\hfill & \text{We}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{parentheses}\text{.}\text{\hspace{0.17em}}\text{They}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{preceded}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{a}\hfill \\ \hfill & \text{"-"}\text{\hspace{0.17em}}\text{sign,}\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{them}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{change}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{each}\hfill \\ \hfill & \text{term}\text{\hspace{0.17em}}\text{inside}\text{\hspace{0.17em}}\text{them}\text{.}\hfill \\ 5{y}^{3}+11-12{y}^{3}+2\hfill & \text{Combine}\text{\hspace{0.17em}}\text{like}\text{\hspace{0.17em}}\text{terms}\text{.}\hfill \\ -7{y}^{3}+13\hfill & \hfill \end{array}$

Add $4{x}^{2}+2x-8$ to $3{x}^{2}-7x-10$ .

$\begin{array}{l}\left(4{x}^{2}+2x-8\right)+\left(3{x}^{2}-7x-10\right)\\ 4{x}^{2}+2x-8+3{x}^{2}-7x-10\\ 7{x}^{2}-5x-18\end{array}$

Subtract $8{x}^{2}-5x+2$ from $3{x}^{2}+x-12$ .

$\begin{array}{l}\left(3{x}^{2}+x-12\right)-\left(8{x}^{2}-5x+2\right)\\ 3{x}^{2}+x-12-8{x}^{2}+5x-2\\ -5{x}^{2}+6x-14\end{array}$

Be very careful not to write this problem as

$3{x}^{2}+x-12-8{x}^{2}-5x+2$

This form has us subtracting only the very first term, $8{x}^{2}$ , rather than the entire expression. Use parentheses.
Another incorrect form is

$8{x}^{2}-5x+2-\left(3{x}^{2}+x-12\right)$

This form has us performing the subtraction in the wrong order.

## Practice set d

Perform the following additions and subtractions.

$6{y}^{2}+2y-1+\left(5{y}^{2}-18\right)$

$11{y}^{2}+2y-19$

$\left(9m-n\right)-\left(10m+12n\right)$

$-m-13n$

Add $2{r}^{2}+4r-1$ to $3{r}^{2}-r-7$ .

$5{r}^{2}+3r-8$

Subtract $4s-3$ from $7s+8$ .

$3s+11$

## Exercises

For the following problems, perform the multiplications and combine any like terms.

$7\left(x+6\right)$

$7x+42$

$4\left(y+3\right)$

$6\left(y+4\right)$

$6y+24$

$8\left(m+7\right)$

$5\left(a-6\right)$

$5a-30$

$2\left(x-10\right)$

$3\left(4x+2\right)$

$12x+6$

$6\left(3x+4\right)$

$9\left(4y-3\right)$

$36y-27$

$5\left(8m-6\right)$

$-9\left(a+7\right)$

$-9a-63$

$-3\left(b+8\right)$

$-4\left(x+2\right)$

$-4x-8$

$-6\left(y+7\right)$

$-3\left(a-6\right)$

$-3a+18$

$-9\left(k-7\right)$

$-5\left(2a+1\right)$

$-10a-5$

$-7\left(4x+2\right)$

$-3\left(10y-6\right)$

$-30y+18$

$-8\left(4y-11\right)$

$x\left(x+6\right)$

${x}^{2}+6x$

$y\left(y+7\right)$

$m\left(m-4\right)$

${m}^{2}-4m$

$k\left(k-11\right)$

$3x\left(x+2\right)$

$3{x}^{2}+6x$

$4y\left(y+7\right)$

$6a\left(a-5\right)$

$6{a}^{2}-30a$

$9x\left(x-3\right)$

$3x\left(5x+4\right)$

$15{x}^{2}+12x$

$4m\left(2m+7\right)$

$2b\left(b-1\right)$

$2{b}^{2}-2b$

$7a\left(a-4\right)$

$3{x}^{2}\left(5{x}^{2}+4\right)$

$15{x}^{4}+12{x}^{2}$

$9{y}^{3}\left(3{y}^{2}+2\right)$

$4{a}^{4}\left(5{a}^{3}+3{a}^{2}+2a\right)$

$20{a}^{7}+12{a}^{6}+8{a}^{5}$

$2{x}^{4}\left(6{x}^{3}-5{x}^{2}-2x+3\right)$

$-5{x}^{2}\left(x+2\right)$

$-5{x}^{3}-10{x}^{2}$

$-6{y}^{3}\left(y+5\right)$

$2{x}^{2}y\left(3{x}^{2}{y}^{2}-6x\right)$

$6{x}^{4}{y}^{3}-12{x}^{3}y$

$8{a}^{3}{b}^{2}c\left(2a{b}^{3}+3b\right)$

${b}^{5}{x}^{2}\left(2bx-11\right)$

$2{b}^{6}{x}^{3}-11{b}^{5}{x}^{2}$

$4x\left(3{x}^{2}-6x+10\right)$

$9{y}^{3}\left(2{y}^{4}-3{y}^{3}+8{y}^{2}+y-6\right)$

$18{y}^{7}-27{y}^{6}+72{y}^{5}+9{y}^{4}-54{y}^{3}$

$-{a}^{2}{b}^{3}\left(6a{b}^{4}+5a{b}^{3}-8{b}^{2}+7b-2\right)$

$\left(a+4\right)\left(a+2\right)$

${a}^{2}+6a+8$

$\left(x+1\right)\left(x+7\right)$

$\left(y+6\right)\left(y-3\right)$

${y}^{2}+3y-18$

$\left(t+8\right)\left(t-2\right)$

$\left(i-3\right)\left(i+5\right)$

${i}^{2}+2i-15$

$\left(x-y\right)\left(2x+y\right)$

$\left(3a-1\right)\left(2a-6\right)$

$6{a}^{2}-20a+6$

$\left(5a-2\right)\left(6a-8\right)$

$\left(6y+11\right)\left(3y+10\right)$

$18{y}^{2}+93y+110$

$\left(2t+6\right)\left(3t+4\right)$

$\left(4+x\right)\left(3-x\right)$

$-{x}^{2}-x+12$

$\left(6+a\right)\left(4+a\right)$

$\left({x}^{2}+2\right)\left(x+1\right)$

${x}^{3}+{x}^{2}+2x+2$

$\left({x}^{2}+5\right)\left(x+4\right)$

$\left(3{x}^{2}-5\right)\left(2{x}^{2}+1\right)$

$6{x}^{4}-7{x}^{2}-5$

$\left(4{a}^{2}{b}^{3}-2a\right)\left(5{a}^{2}b-3b\right)$

$\left(6{x}^{3}{y}^{4}+6x\right)\left(2{x}^{2}{y}^{3}+5y\right)$

$12{x}^{5}{y}^{7}+30{x}^{3}{y}^{5}+12{x}^{3}{y}^{3}+30xy$

$5\left(x-7\right)\left(x-3\right)$

$4\left(a+1\right)\left(a-8\right)$

$4{a}^{2}-28a-32$

$a\left(a-3\right)\left(a+5\right)$

$x\left(x+1\right)\left(x+4\right)$

${x}^{3}+5{x}^{2}+4x$

${x}^{2}\left(x+5\right)\left(x+7\right)$

${y}^{3}\left(y-3\right)\left(y-2\right)$

${y}^{5}-5{y}^{4}+6{y}^{3}$

$2{a}^{2}\left(a+4\right)\left(a+3\right)$

$5{y}^{6}\left(y+7\right)\left(y+1\right)$

$5{y}^{8}+40{y}^{7}+35{y}^{6}$

$a{b}^{2}\left({a}^{2}-2b\right)\left(a+{b}^{4}\right)$

${x}^{3}{y}^{2}\left(5{x}^{2}{y}^{2}-3\right)\left(2xy-1\right)$

$10{x}^{6}{y}^{5}-5{x}^{5}{y}^{4}-6{x}^{4}{y}^{3}+3{x}^{3}{y}^{2}$

$6\left({a}^{2}+5a+3\right)$

$8\left({c}^{3}+5c+11\right)$

$8{c}^{3}+40c+88$

$3{a}^{2}\left(2{a}^{3}-10{a}^{2}-4a+9\right)$

$6{a}^{3}{b}^{3}\left(4{a}^{2}{b}^{6}+7a{b}^{8}+2{b}^{10}+14\right)$

$24{a}^{5}{b}^{9}+42{a}^{4}{b}^{11}+12{a}^{3}{b}^{13}+18{a}^{3}{b}^{3}$

$\left(a-4\right)\left({a}^{2}+a-5\right)$

$\left(x-7\right)\left({x}^{2}+x-3\right)$

${x}^{3}-6{x}^{2}-10x+21$

$\left(2x+1\right)\left(5{x}^{3}+6{x}^{2}+8\right)$

$\left(7{a}^{2}+2\right)\left(3{a}^{5}-4{a}^{3}-a-1\right)$

$21{a}^{7}-22{a}^{5}-15{a}^{3}-7{a}^{2}-2a-2$

$\left(x+y\right)\left(2{x}^{2}+3xy+5{y}^{2}\right)$

$\left(2a+b\right)\left(5{a}^{2}+4{a}^{2}b-b-4\right)$

$10{a}^{3}+8{a}^{3}b+4{a}^{2}{b}^{2}+5{a}^{2}b-{b}^{2}-8a-4b-2ab$

${\left(x+3\right)}^{2}$

${\left(x+1\right)}^{2}$

${x}^{2}+2x+1$

${\left(x-5\right)}^{2}$

${\left(a+2\right)}^{2}$

${a}^{2}+4a+4$

${\left(a-9\right)}^{2}$

$-{\left(3x-5\right)}^{2}$

$-9{x}^{2}+30x-25$

$-{\left(8t+7\right)}^{2}$

For the following problems, perform the indicated operations and combine like terms.

$3{x}^{2}+5x-2+\left(4{x}^{2}-10x-5\right)$

$7{x}^{2}-5x-7$

$-2{x}^{3}+4{x}^{2}+5x-8+\left({x}^{3}-3{x}^{2}-11x+1\right)$

$-5x-12xy+4{y}^{2}+\left(-7x+7xy-2{y}^{2}\right)$

$2{y}^{2}-5xy-12x$

$\left(6{a}^{2}-3a+7\right)-4{a}^{2}+2a-8$

$\left(5{x}^{2}-24x-15\right)+{x}^{2}-9x+14$

$6{x}^{2}-33x-1$

$\left(3{x}^{3}-7{x}^{2}+2\right)+\left({x}^{3}+6\right)$

$\left(9{a}^{2}b-3ab+12a{b}^{2}\right)+a{b}^{2}+2ab$

$9{a}^{2}b+13a{b}^{2}-ab$

$6{x}^{2}-12x+\left(4{x}^{2}-3x-1\right)+4{x}^{2}-10x-4$

$5{a}^{3}-2a-26+\left(4{a}^{3}-11{a}^{2}+2a\right)-7a+8{a}^{3}+20$

$17{a}^{3}-11{a}^{2}-7a-6$

$2xy-15-\left(5xy+4\right)$

Add $4x+6$ to $8x-15$ .

$12x-9$

Add $5{y}^{2}-5y+1$ to $-9{y}^{2}+4y-2$ .

Add $3\left(x+6\right)$ to $4\left(x-7\right)$ .

$7x-10$

Add $-2\left({x}^{2}-4\right)$ to $5\left({x}^{2}+3x-1\right)$ .

Add four times $5x+2$ to three times $2x-1$ .

$26x+5$

Add five times $-3x+2$ to seven times $4x+3$ .

Add $-4$ times $9x+6$ to $-2$ times $-8x-3$ .

$-20x-18$

Subtract $6{x}^{2}-10x+4$ from $3{x}^{2}-2x+5$ .

Substract ${a}^{2}-16$ from ${a}^{2}-16$ .

0

## Exercises for review

( [link] ) Simplify ${\left(\frac{15{x}^{2}{y}^{6}}{5x{y}^{2}}\right)}^{4}$ .

( [link] ) Express the number 198,000 using scientific notation.

$1.98×{10}^{5}$

( [link] ) How many $4{a}^{2}{x}^{3}\text{'}\text{s}$ are there in $-16{a}^{4}{x}^{5}$ ?

( [link] ) State the degree of the polynomial $4x{y}^{3}+3{x}^{5}y-5{x}^{3}{y}^{3}$ , and write the numerical coefficient of each term.

$\text{degree}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}6;\text{\hspace{0.17em}}\text{\hspace{0.17em}}4,3,-5$

( [link] ) Simplify $3\left(4x-5\right)+2\left(5x-2\right)-\left(x-3\right)$ .

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