Lesson 5: Polynomial Long Division with Remainders Topic Solutions

When a solution has a remainder, write it as the fraction: $\frac{remainder}{divisor}$
Add the remainder to the quotient for the complete solution.

The remainder is a rational expression.

Some polynomials in standard form have a missing degree.
For example, $x^{2} - 1,$ is missing the first degree (linear) term.
It must be rewritten as $x^{2} + 0 x - 1$ using the Additive Identity Property .
Recall that all constant terms (numbers) are 0 degree terms since $x^{0} = 1 .$
If any degree term is “missing,” write it as $0 x^{n}$ where n is the missing degree.
Use the same long division process for finding the quotient once you write all of the terms in descending order.

Example 7

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

Plan
Write the expressions with the long division symbol

Simplify from largest to smaller degree terms using the divisor

Write the quotient adhering to place-value by degree

Write remainder as a rational expression

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

$x + 1 2 x^{2} - 3 x + 4 2 x^{3} - x^{2} + x + 1 - (2 x^{3} + 2 x^{2}) - 3 x^{2} + x - (- 3 x^{2} - 3 x) 4 x + 1 - (4 x + 4) - 3 - \frac{3}{x + 1}$

Note

Find the value that can be used to eliminate $2 x^{3}$
$(2 x^{2}) (x + 1) = 2 x^{3} + 2 x^{2}$
Subtract
Find the value that can be used to eliminate $- 3 x^{2}$
$(- 3 x) (x + 1) = - 3 x^{2} - 3 x$
Subtract
Find the value that can be used to eliminate + 4x
$(4) (x + 1) = 4 x + 4$
Subtract
The remainder is –3

Check

$(x + 1) (2 x^{2} - 3 x + 4) - 3 2 x^{3} - 3 x^{2} + 4 x + 2 x^{2} - 3 x + 4 - 3 2 x^{3} - x^{2} + x + 1 ✓$

Example 8

Simplify.

$(8 x^{3} - 4 x - \frac{11}{8}) \div (4 x - 3)$

$4 x - 3 2 x^{2} + \frac{3}{2} x + \frac{1}{8} 8 x^{3} + 0 x^{2} - 4 x - \frac{11}{8} - (8 x^{3} - 6 x^{2}) 6 x^{2} - \frac{8}{2} x - (\frac{}{} 6 x^{2} - \frac{9}{2} x) \frac{1}{2} x - \frac{11}{8} - (\frac{1}{2} x - \frac{3}{8}) - 1 - \frac{1}{4 x - 3}$

Note

A common denominator is needed for this problem to be able to correctly combine like terms.

Check

$(4 x - 3) (2 x^{2} + \frac{3}{2} x + \frac{1}{8}) - 1 8 x^{3} + 6 x^{2} + \frac{1}{2} x - 6 x^{2} - \frac{9}{2} x - \frac{11}{8} ✓$

Note

Refer to the Lesson 6 More to Explore on Polydoku division to see an alternate method to long division.