# Dividing Polynomial Functions

You can divide a polynomial just like numbers and dividing a polynomial will give an expression as quotient and remainder. The polynomial you are going to divide must have more terms than the divisor, else the division will not be fruitful.

### Polynomial Long Division

To divide a polynomial, you need to follow some steps.

Example #1

Step 1: Keep the terms of the dividend and divisor in standard form, that is, in the descending powers of the variable.

x + 2 )\overline {x^2 + 7x + 10}

Step 2: Take first term of the dividend and divide with the divisor, you will get the first term of the quotient. Do it as if you are dividing two numbers.

x\\
x + 2 )\overline {x^2 + 7x + 10}

Step 3: Multiply every term of the divisor with the first term of the quotient.

x\\
x + 2 )\overline{x^2 + 7x + 10}\\
x^2 + 2x 

Step 4: Subtract the result of multiplication in step 3 with dividend.

x\\
x + 2 ) \overline{x^2 + 7x + 10} \\
x^2 + 2x\\
\hspace{1.5 cm}\overline{5x + 10} 

Step 5: Bring down the new term from the dividend and repeat the step 1 to 5 until you get a remainder 0 or some other value.

x + 5\\
x + 2 ) \overline{x^2 + 7x + 10} \\
x^2 + 2x\\
\hspace{2 cm}\overline{5x + 10} \\
\hspace{2 cm}5x + 10\\
\hspace{3 cm}\overline{0}\\

The solution to the above polynomial is which is you obtained after the polynomial long division as quotient. Let perform a polynomial long division on another polynomial.

Example #2:

Divide \hspace{3 mm} x - 1 ) \overline{x^3 + 3x^2 + 3x -4}

Solution:

\begin{aligned}
&\hspace{ 1 cm}x^2 + 4x + 7 \\
&x - 1 ) \overline{x^3 + 3x^2 + 3x - 4} \\
&\hspace{ 1 cm}x^3 - x^2  \\
&\hspace{ 1 cm}\overline{4x^2 + 3x } \\
&\hspace{ 1 cm}4x^2 - 4x\\
&\hspace{ 2 cm}\overline{7x - 4} \\
&\hspace{ 2 cm}7x - 7\\
&\hspace{ 2.7 cm}\overline{11}\\
\end{aligned}

This time the solution is a trinomial . Note that the result of polynomial long division is not always zero, you may get a non-zero remainder too.

How do you write answer when the polynomial long division gives you a remainder. As in above case, you can write answers as

\frac{x^3 + 3x^2 + 3x -4}{x - 1} = x^2 + 4x + 7 + \frac{13}{x - 1}

The divisor still tries to divide the remainder so you represent it as a term.

### Division Algorithm

We can rewrite the whole dividend , divisor , quotient and remainder as an expression by itself.

x^3 + 3x^2 + 3x -4 = (x - 1)(x^2 + 4x + 7) +11

What we are doing is a check to see whether multiplying and adding the remainder back will give us the original polynomial function

Let say that the dividend is , divisor is , quotient is and remainder is .

The degree of divisor is less than or equal to , where . Also, there exists unique polynomials and .

The degree of remainder is 0 or less than degree of divisor . If remainder , then we can say that divisor divides polynomial evenly and and are factors of the polynomial.

### Synthetic Division

Another way to divide polynomial is much faster and efficient provide that the divisor is in the form where is a constant.

We can take our previous example to perform the synthetic division.

Example #3

Divide using synthetic division method:

Solution:

Step 1: Write the constant $c$ from the divisor and all the coefficients from the dividend.

Step 3: Write the 1st term coefficient in third column.

Step 4: Multiply 1st term coefficient with c and write the result in second column and second row and add the entries of second column. Write the result in second column, third row.

Step 5: Repeat the process from step 1 to 4.

The numbers in the last row is coefficients of quotient and remainder . Therefore, the result is

q(x) = x^2 + 4x + 7 + \frac{11}{x -1}

### Remainder Theorem

Let us remember the division algorithm, which is

f(x) = d(x) . q(x) + r

Suppose we are dividing with , then remainder will be a constant , which means,

f(x) = (x - c) q(x) + r

Given the above equation, suppose , then

f(c) = (c - c)q(c) + r\\
= 0 \cdot q(c) + r\\
f(x) = r



Therefore, remainder theorem states that you can use from and get the remainder.

Example #4

Find the remainder for divided by

Solution:

Using reminder theorem, and .

f(1) = 1^3 + 4(1)^2 + 3(1) + 2 = 1 + 4 + 3 + 2 = 10

Therefore, remainder is .

Verify the results:

1 : 1  4  3  2
1  5  8
1  5  8  10

Using the synthetic division we find that the remainder is 10.

### Factor Theorem

The factor theorem is derived from division algorithm. Suppose the divides the polynomial . We know that by remainder theorem, which result in .

Let us replace in above equation as . Now the division algorithm becomes .

Suppose if the then, the equation becomes which implies that is a factor of .

Let us say that is a factor of polynomial .Then,

f(x) = (x - c) q(x) ,  if \hspace{2 mm} x = c\\
f(c) = (c - c) q(c) = 0 \cdot q(c) = 0

If is a factor of , then . This is known as factor theorem.

### Summary

In this article, you learned about polynomial long division, synthetic division, division algorithm and two theorem that are derived from the division algorithms – remainder theorem and factor theorem.