Dividing Polynomial Expressions
What is a Polynomial?
A polynomial is an expression obtained when we add or subtract two or more monomials.
Dividing Polynomials
Dividing polynomials with monomial expression is easy, all you have to do is divide each term of the polynomial by the monomial.
Example 1: Dividing $\displaystyle {10x+5}$ by $\displaystyle 5$.
$\displaystyle \frac{{10x+5}}{5}=\frac{{10x}}{5}+\frac{5}{5}=2x+1$
Example 2 :Dividing $\displaystyle \frac{{12{{x}^{2}}+8x+20}}{{4x}}=$
$\displaystyle \frac{{12{{x}^{2}}}}{{4x}}+\frac{{8x}}{{4x}}+\frac{{20}}{{4x}}=$$\displaystyle 3x+2+\frac{5}{x}$
Dividing polynomials with polynomial algebraic expressions is a little bit more complicated.
We use two methods for dividing polynomials
The Long Division Method and Synthetic Method
The division method is a method similar to long division of numbers.
How to divide a polynomial using long division method:
Step 1: Write the polynomial on the descending order. Use zero in the place of the missing terms.
Step 2: Divide the term of the highest power of the polynomial with the term of the highest power of the divisor.
Step 3: Multiply the answer obtained in step 2 with the divisor.
Step 4: Subtract and bring down the next term.
Step 5: Repeat the steps above till there are no more terms to bring down.
Step 6: Write the final answer and don’t forget to write the remainder as a fraction if you have a last term.
Example 3: Divide $\displaystyle {{x}^{3}}+10{{x}^{2}}+11x-70$ by $\displaystyle x-2$.
The answer: $\displaystyle \frac{{{{x}^{3}}+10{{x}^{2}}+11x-70}}{{x-2}}=$$ \displaystyle {{x}^{2}}+12x+35$
Example 4: Divide $\displaystyle 2{{x}^{4}}+7{{x}^{3}}-14{{x}^{2}}-3x+15$ by $\displaystyle x+5$.
The answer: $\displaystyle \frac{{2{{x}^{4}}+7{{x}^{3}}-14{{x}^{2}}-3x+15}}{{x+5}}=$$\displaystyle 2{{x}^{3}}-3{{x}^{2}}+x-8+\frac{{55}}{{x+5}}$
The answer: $\displaystyle \frac{{{{x}^{4}}+{{x}^{2}}+x+3}}{{{{x}^{2}}-1}}=$$ \displaystyle {{x}^{2}}+2+\frac{{x+5}}{{{{x}^{2}}-1}}$
Another method to divide polynomials is the synthetic method (also known as Ruffini’s rule) which is a more simple way to divide a polynomial by a first degree polynomial with leading coefficient 1. This method uses a table filled with coefficients of polynomials for better understanding.
How to divide a polynomial using Synthetic Method
Step 1: Set the divisor to zero to find the number to put in the division box, write the numerator in the descending order and fill with zero the missing terms when written the coefficients on the table.
Step 2: Bring the leading coefficient straight down two times.
Step 3: Multiply the number in the division box with the number you brought down and put the result on the next column and add the two numbers together and write the result on the next row.
Step 4: Multiply the number in the division box with the result you wrote on step 3 and put the result on the next column and add the two numbers together and write the result on the next row.
Step 5: Multiply the number in the division box with the result you wrote on step 4 and put the result on the next column and add the two numbers together and write the result on the next row.
Step 6: Write the final answer, which is made from the coefficients on the last row with last number the reminder and the power of variables x-s are one power less than the numerator. Don’t forget to write the reminder as a fraction.
Example 6: Divide $\displaystyle {{x}^{3}}+2{{x}^{2}}-4x-3$ by$\displaystyle x+3$.
Step 6:
Since the last number on the table is 0 we don’t have a remainder.
The Answer: $\displaystyle \frac{{{{x}^{3}}+2{{x}^{2}}-4x-3}}{{x+3}}=$$\displaystyle {{x}^{2}}+x-1$
Example 7: Divide $ \displaystyle {{x}^{3}}-2x+1$ by $\displaystyle x-1$.
The Answer: $\displaystyle \frac{{{{x}^{3}}-2x+1}}{{x-1}}=$$\displaystyle {{x}^{2}}+x-1$
Example 8: Divide $\displaystyle 2{{x}^{4}}-17{{x}^{3}}+22{{x}^{2}}+65x-9$ by $\displaystyle x-9$.
The last number on the table is the remainder.
The answer: $\displaystyle \frac{{2{{x}^{4}}-17{{x}^{3}}+22{{x}^{2}}+65x-9}}{{x-9}}=$ $\displaystyle 2{{x}^{3}}+{{x}^{2}}+31x+334+\frac{{3087}}{{x-9}}$
