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The process was to divide, multiply, and subtract.
A very important step in the process of dividing one polynomial by another is subtraction of polynomials. Let’s review the process of subtraction by observing a few examples.
1. Subtract $x-2$ from $x-5;$ that is, find $\left(x-5\right)-\left(x-2\right).$
Since $\mathrm{\hspace{0.17em}\hspace{0.17em}x}-2$ is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.
$\frac{\begin{array}{l}\mathrm{\hspace{0.17em}\hspace{0.17em}x}-5\\ -\left(x-2\right)\end{array}}{}=\frac{\begin{array}{l}x-5\\ -x+2\end{array}}{-3}$
The result is $-3.$
2. Subtract ${x}^{3}+3{x}^{2}$ from ${x}^{3}+4{x}^{2}+x-1.$
Since ${x}^{3}+3{x}^{2}$ is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.
$\frac{\begin{array}{l}{x}^{3}+4{x}^{2}+x-1\\ -\left({x}^{3}+3{x}^{2}\right)\end{array}}{}=\frac{\begin{array}{l}{x}^{3}+4{x}^{2}+x-1\\ -{x}^{3}-3{x}^{2}\end{array}}{{x}^{2}+x-1}$
The result is ${\text{x}}^{2}+x-1.$
3. Subtract ${x}^{2}+3x$ from ${x}^{2}+1.$
We can write ${x}^{2}+1$ as ${x}^{2}+0x+1.$
$\frac{\begin{array}{l}{x}^{2}+1\\ -\left({x}^{2}+3x\right)\end{array}}{}=\frac{\begin{array}{l}{x}^{2}+0x+1\\ -\left({x}^{2}+3x\right)\end{array}}{}=\frac{\begin{array}{l}{x}^{2}+0x+1\\ -{x}^{2}-3x\end{array}}{-3x+1}$
Now we’ll observe some examples of dividing one polynomial by another. The process is the same as the process used with whole numbers: divide, multiply, subtract, divide, multiply, subtract,....
The division, multiplication, and subtraction take place one term at a time. The process is concluded when the polynomial remainder is of lesser degree than the polynomial divisor.
Perform the division.
$\begin{array}{lll}\frac{x-5}{x-2}.\hfill & \hfill & \text{We\hspace{0.17em}are\hspace{0.17em}to\hspace{0.17em}divide\hspace{0.17em}}x-5\text{\hspace{0.17em}by\hspace{0.17em}}x-2.\hfill \end{array}$
$\begin{array}{l}1-\frac{3}{x-2}\\ \text{Thus,}\\ \frac{x-5}{x-2}=1-\frac{3}{x-2}\end{array}$
$\begin{array}{lll}\frac{{x}^{3}+4{x}^{2}+x-1}{x+3}.\hfill & \hfill & \text{We\hspace{0.17em}are\hspace{0.17em}to\hspace{0.17em}divide\hspace{0.17em}}{x}^{3}+4{x}^{2}+x-1\text{\hspace{0.17em}by\hspace{0.17em}}x+3.\hfill \end{array}$
$\begin{array}{l}{x}^{2}+x-2+\frac{5}{x+3}\\ \text{Thus,}\\ \frac{{x}^{3}+4{x}^{2}+x-1}{x+3}={x}^{2}+x-2+\frac{5}{x+3}\end{array}$
Perform the following divisions.
$\frac{{x}^{3}+{x}^{2}-x-2}{x+8}$
${x}^{2}-7x+55-\frac{442}{x+8}$
$\frac{{x}^{3}+{x}^{2}-3x+1}{{x}^{2}+4x-5}$
$x-3+\frac{14x-14}{{x}^{2}+4x-5}=x-3+\frac{14}{x+5}$
$\begin{array}{l}\text{Divide\hspace{0.17em}}2{x}^{3}-4x+1\text{\hspace{0.17em}}\text{by\hspace{0.17em}}x+6.\\ \begin{array}{lll}\frac{2{x}^{3}-4x+1}{x+6}\hfill & \hfill & \begin{array}{l}\text{Notice\hspace{0.17em}that\hspace{0.17em}the\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}term\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}is\hspace{0.17em}missing}\text{.\hspace{0.17em}}\\ \text{We\hspace{0.17em}can\hspace{0.17em}avoid\hspace{0.17em}any\hspace{0.17em}confusion\hspace{0.17em}by\hspace{0.17em}writing}\end{array}\hfill \\ \frac{2{x}^{3}+0{x}^{2}-4x+1}{x+6}\hfill & \hfill & \text{Divide,\hspace{0.17em}multiply,\hspace{0.17em}and\hspace{0.17em}subtract}\text{.}\hfill \end{array}\end{array}$
$\frac{2{x}^{3}-4x+1}{x+6}=2{x}^{3}-12x+68-\frac{407}{x+6}$
Perform the following divisions.
$\frac{6{x}^{3}+5{x}^{2}-1}{2x+3}$
$3{x}^{2}-2x+3-\frac{10}{2x+3}$
For the following problems, perform the divisions.
$\frac{12b-6}{3}$
$\frac{21a-9}{-3}$
$\frac{4{y}^{2}-2y}{2y}$
$\frac{20{x}^{2}+10x}{5x}$
$\frac{26{y}^{3}+13{y}^{2}+39y}{13y}$
$\frac{{a}^{2}{b}^{2}+4{a}^{2}b+6a{b}^{2}-10ab}{ab}$
$ab+4a+6b-10$
$\frac{7{x}^{3}y+8{x}^{2}{y}^{3}+3x{y}^{4}-4xy}{xy}$
$\frac{5{x}^{3}{y}^{3}-15{x}^{2}{y}^{2}+20xy}{-5xy}$
$-{x}^{2}{y}^{2}+3xy-4$
$\frac{4{a}^{2}{b}^{3}-8a{b}^{4}+12a{b}^{2}}{-2a{b}^{2}}$
$\frac{6{a}^{2}{y}^{2}+12{a}^{2}y+18{a}^{2}}{24{a}^{2}}$
$\frac{1}{4}{y}^{2}+\frac{1}{2}y+\frac{3}{4}$
$\frac{3{c}^{3}{y}^{3}+99{c}^{3}{y}^{4}-12{c}^{3}{y}^{5}}{3{c}^{3}{y}^{3}}$
$\frac{16a{x}^{2}-20a{x}^{3}+24a{x}^{4}}{6{a}^{4}}$
$\frac{8{x}^{2}-10{x}^{3}+12{x}^{4}}{3{a}^{3}}\text{or}\frac{12{x}^{4}-10{x}^{3}+8{x}^{2}}{3{a}^{3}}$
$\frac{21a{y}^{3}-18a{y}^{2}-15ay}{6a{y}^{2}}$
$\frac{-14{b}^{2}{c}^{2}+21{b}^{3}{c}^{3}-28{c}^{3}}{-7{a}^{2}{c}^{3}}$
$\frac{2{b}^{2}-3{b}^{3}c+4c}{{a}^{2}c}$
$\frac{-30{a}^{2}{b}^{4}-35{a}^{2}{b}^{3}-25{a}^{2}}{-5{b}^{3}}$
$\frac{y+7}{y+1}$
$\frac{{x}^{2}+2x-1}{x+1}$
$\frac{{x}^{2}+5x+5}{x+5}$
$\frac{{a}^{2}-6}{a+2}$
$\frac{{x}^{2}+36}{x+6}$
$\frac{{a}^{3}-8}{a+2}$
$\frac{{a}^{3}-8}{a-2}$
$\frac{{x}^{3}+3{x}^{2}+x-2}{x-2}$
${x}^{2}+5x+11+\frac{20}{x-2}$
$\frac{{a}^{3}+2{a}^{2}-a+1}{a-3}$
$\frac{{x}^{3}+2x+1}{x-3}$
$\frac{{y}^{3}+3{y}^{2}+4}{y+2}$
${y}^{2}+y-2+\frac{8}{y+2}$
$\frac{{y}^{3}+5{y}^{2}-3}{y-1}$
$\frac{{a}^{2}+2a}{a+2}$
$\frac{{a}^{2}+5a+4}{{a}^{2}-a-2}$
$\frac{3{a}^{2}+4a-4}{{a}^{2}+3a+3}$
$\frac{3{a}^{2}+4a+2}{3a+4}$
$\frac{20{y}^{2}+15y-4}{4y+3}$
$\frac{4{x}^{3}+4{x}^{2}-3x-2}{2x-1}$
$2{x}^{2}+3x-\frac{2}{2x-1}$
$\frac{9{a}^{3}-18{a}^{2}+8a-1}{3a-2}$
$\frac{4{x}^{4}-4{x}^{3}+2{x}^{2}-2x-1}{x-1}$
$4{x}^{3}+2x-\frac{1}{x-1}$
$\frac{3{y}^{4}+9{y}^{3}-2{y}^{2}-6y+4}{y+3}$
$\frac{3{y}^{2}+3y+5}{{y}^{2}+y+1}$
$3+\frac{2}{{y}^{2}+y+1}$
$\frac{2{a}^{2}+4a+1}{{a}^{2}+2a+3}$
$\frac{8{z}^{6}-4{z}^{5}-8{z}^{4}+8{z}^{3}+3{z}^{2}-14z}{2z-3}$
$4{z}^{5}+4{z}^{4}+2{z}^{3}+7{z}^{2}+12z+11+\frac{33}{2z-3}$
$\frac{9{a}^{7}+15{a}^{6}+4{a}^{5}-3{a}^{4}-{a}^{3}+12{a}^{2}+a-5}{3a+1}$
$(2{x}^{5}+5{x}^{4}-1)\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}(2x+5)$
${x}^{4}-\frac{1}{2x+5}$
$(6{a}^{4}-2{a}^{3}-3{a}^{2}+a+4)\text{\hspace{0.17em}}\xf7\text{\hspace{0.17em}}(3a-1)$
( [link] ) Find the product. $\frac{{x}^{2}+2x-8}{{x}^{2}-9}\xb7\frac{2x+6}{4x-8}.$
$\frac{x+4}{2\left(x-3\right)}$
( [link] ) Find the sum. $\frac{x-7}{x+5}+\frac{x+4}{x-2}.$
( [link] ) Solve the equation $\frac{1}{x+3}+\frac{1}{x-3}=\frac{1}{{x}^{2}-9}.$
$x=\frac{1}{2}$
( [link] ) When the same number is subtracted from both the numerator and denominator of $\frac{3}{10}$ , the result is $\frac{1}{8}$ . What is the number that is subtracted?
( [link] ) Simplify $\frac{\frac{1}{x+5}}{\frac{4}{{x}^{2}-25}}.$
$\frac{x-5}{4}$
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