나머지정리와 인수정리

When we divide monomials with more than one variable, we write one fraction for each variable.

$-\frac{9}{{a}^{5}b}$

$\frac{-9d}{{c}^{4}}$

Be very careful to simplify $\frac{14}{21}$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

$\frac{4{y}^{2}}{7{x}^{4}}$

$\frac{5}{8{m}^{5}{n}^{3}}$

$-4a+2b$

$8{a}^{5}b+6{a}^{3}{b}^{2}$

$y+7$

$m+5$

Notice that there is no ${x}^{3}$ term in the dividend. We will add $0{x}^{3}$ as a placeholder.

Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.
Divide ${x}^{4}$ by $x.$ Put the answer, ${x}^{3},$ in the quotient over the ${x}^{3}$ term. Multiply ${x}^{3}$ times $x+2.$ Line up the like terms. Subtract and then bring down the next term.
Divide $-2{x}^{3}$ by $x.$ Put the answer, $-2{x}^{2},$ in the quotient over the ${x}^{2}$ term. Multiply $-2{x}^{2}$ times $x+1.$ Line up the like terms Subtract and bring down the next term.
Divide $3{x}^{2}$ by $x.$ Put the answer, $3x,$ in the quotient over the $x$ term. Multiply $3x$ times $x+1.$ Line up the like terms. Subtract and bring down the next term.
Divide $-x$ by $x.$ Put the answer, $-1,$ in the quotient over the constant term. Multiply $-1$ times $x+1.$ Line up the like terms. Change the signs, add. Write the remainder as a fraction with the divisor as the denominator.
To check, multiply $(x+2)\left({x}^{3}-2{x}^{2}+3x-1-\frac{4}{x+2}\right)$. The result should be ${x}^{4}-{x}^{2}+5x-6.$

${x}^{3}-3{x}^{2}+2x+1+\frac{3}{x+3}$

${x}^{3}-3{x}^{2}-2x-1-\frac{3}{x+3}$

This time we will show the division all in one step. We need to add two placeholders in order to divide.

To check, multiply $(2a+3)(4{a}^{2}-6a+9).$

The result should be $8{a}^{3}+27.$

${x}^{2}+4x+16$

$25{x}^{2}+10x+4$

Write the dividend with decreasing powers of $x.$
Write the coefficients of the terms as the first row of the synthetic division.
Write the divisor as $x-c$ and place c in the synthetic division in the divisor box.
Bring down the first coefficient to the third row.
Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Add the second column, putting the result in the third row.
Multiply that result by the divisor and place the result in the second row under the third coefficient.
Add the third column, putting the result in the third row.
Multiply that result by the divisor and place the result in the third row under the third coefficient.
Add the final column, putting the result in the third row.
The quotient is $2{x}^{2}-1x+3$ and the remainder is 2.

The division is complete. The numbers in the third row give us the result. The $2\quad -1\quad 3$ are the coefficients of the quotient. The quotient is $2{x}^{2}-1x+3.$ The 2 in the box in the third row is the remainder.

Check:

$\quad \begin{array}{ccc} \\ \\ \text{(quotient)(divisor)}+\text{remainder} & = & \text{dividend} \\ (2{x}^{2}-1x+3)(x+2)+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}-{x}^{2}+3x+4{x}^{2}-2x+6+2 & \overset{?}{=} & 2{x}^{3}+3{x}^{2}+x+8 \\ 2{x}^{3}+3{x}^{2}+x+8 & = & 2{x}^{3}+3{x}^{2}+x+8\checkmark \end{array}$

$3{x}^{2}+4x-2;\,2$