Dividing by Monomials
Warm Up #1: $\displaystyle \frac{24x^5}{8x^2}$
Warm Up #2: Quotient and remainder for the division problem $3\overline{)137}$
Quotient: $\color{blue}{45}$
Remainder: $\color{red}{2}$
This means that $$137=3\cdot \color{blue}{45}+\color{red}{2}$$ or, that from $137,$ we can make $\color{blue}{45}$ groups of $3$ with $\color{red}{2}$ left over.
Equivalently, we may also write $$ \frac{137}{3}=\color{blue}{45}+\frac{\color{red}{2}}{3}=45\frac{2}{3} $$
Remainder: $\color{red}{2}$
This means that $$137=3\cdot \color{blue}{45}+\color{red}{2}$$ or, that from $137,$ we can make $\color{blue}{45}$ groups of $3$ with $\color{red}{2}$ left over.
Equivalently, we may also write $$ \frac{137}{3}=\color{blue}{45}+\frac{\color{red}{2}}{3}=45\frac{2}{3} $$
Example: $\displaystyle \frac{24m^4 - 18m^3 + 36m^2 - 6m}{3m}$
$$
\begin{array}{lll}
\displaystyle \frac{24m^4 - 18m^3 + 36m^2 - 6m}{3m}&\displaystyle=\frac{1}{3m}\left(24m^4 - 18m^3 + 36m^2 - 6m\right) &\mbox{multiply by reciprocal}\\
\displaystyle &\displaystyle=\frac{24m^4}{3m} - \frac{18m^3}{3m} + \frac{36m^2}{3m} -\frac{6m}{3m} &\mbox{distribute (some might skip previous step)}\\
\displaystyle &\displaystyle=8m^3 - 6m^2 + 12m -2 &\mbox{simplify each fraction}\\
\end{array}
$$
General Polynomial Division
- Step 1. Write the polynomials in long-division format, expressing each in standard form.
- Step 2. Divide the first term of the divisor into the first term of the dividend. The result is the first term of the quotient.
- Step 3. Multiply the first term of the quotient by every term in the divisor, and write this product under the dividend, aligning like terms.
- Step 4. Subtract this product from the dividend, and bring down the next term.
- Step 5. Use the result of step 4 as a new dividend, and repeat steps 2 through 4 until either the remainder is 0 or the degree of the remainder is less than the degree of the divisor.
Examples
Perform polynomial long division to get the quotient and remainder. $$2x+7 \,\, \overline{)\,\,\,8x^3+22x^2-23x-11\,\,\,\,\,}$$
Quotient: $\color{blue}{4x^2-3x-1}$
Remainder: $\color{red}{-4}$
This means that $$8x^3+22x^2-23x-11=(2x+7)(\color{blue}{4x^2-3x-1})+\color{red}{(-4)}$$ Or equivalently, $$ \frac{8x^3+22x^2-23x-11}{2x+7}=\color{blue}{4x^2-3x-1}+\frac{\color{red}{-4}}{2x+7} $$
One of the big reasons polynomial long division is useful is that in many cases, the version on the right is easier to work with than the one on the left. This is why we learn it.
Remainder: $\color{red}{-4}$
This means that $$8x^3+22x^2-23x-11=(2x+7)(\color{blue}{4x^2-3x-1})+\color{red}{(-4)}$$ Or equivalently, $$ \frac{8x^3+22x^2-23x-11}{2x+7}=\color{blue}{4x^2-3x-1}+\frac{\color{red}{-4}}{2x+7} $$
One of the big reasons polynomial long division is useful is that in many cases, the version on the right is easier to work with than the one on the left. This is why we learn it.
Quotient: $-4x-9$
Remainder: $6$
Remainder: $6$
Quotient: $-4x+6$
Remainder: $-9x+1$
Remainder: $-9x+1$
Quotient: $-9x+6$
Remainder: $8$
Remainder: $8$