代数学:
多項式除算

1.  
x3−12x2−42x−3\frac{{x}^{3}-12{x}^{2}-42}{x-3}
  解答
2.  
x3−12x2−42x2+x−3\frac{{x}^{3}-12{x}^{2}-42}{{x}^{2}+x-3}
  解答
3.  
x3−1x+2\frac{{x}^{3}-1}{x+2}
  解答
4.  
3x3−2x2−3+4x3+3x+x2\frac{3{x}^{3}-2{x}^{2}-3+4x}{3+3x+{x}^{2}}
  解答
5.  
x3+4x2−9x+3\frac{{x}^{3}+4{x}^{2}-9}{x+3}
  解答
6.  
x3+5x+2x2+10x+2\frac{{x}^{3}+5x+2{x}^{2}+10}{x+2}
  解答
7.  
x3−4x+1x2−4\frac{{x}^{3}-4x+1}{{x}^{2}-4}
  解答
8.  
u24+u\frac{{u}^{2}}{4+u}
  解答

Polynomial Division - Introduction

When you see a polynomial that is a fraction composed of two polynomials — one as the numerator and the other as the denominator — it can often be simplified using long division. The procedure is based on the same principle of long division for whole numbers. Here is how it works.

Recap on Polynomials

The word "polynomial" means "many terms" — something like
4x+2y−34x+2y-3
is a common polynomial form. Note that while polynomials can contain constants (such as
55
or
−17-17
), variables (such as
xx
and
yy
) and exponents, such as
x2{x}^{2}
or
x3{x}^{3}
, polynomials must not contain negative exponents or division by a variable such as
2x\frac{2}{x}
.

An Example

Let’s try some polynomial division practice. Consider this polynomial:
x3−1x+2\frac{{x}^{3}-1}{x+2}
First, we rewrite this as a form of long division. The only difference from regular long divisions is that, instead of numbers, they are polynomials.
Step 1: Divide
x3{x}^{3}
by
xx
, which gives
x2{x}^{2}
. Write this on the line above
x3{x}^{3}
.
Step 2: Multiply
x2{x}^{2}
by
x+2x+2
, which gives
x3+2x2{x}^{3}+2{x}^{2}
. Write this below
x3{x}^{3}
.
Step 3: Subtract
x3+2x2{x}^{3}+2{x}^{2}
from
x3{x}^{3}
to give
−2x2-2{x}^{2}
. Write this below
x3+2x2{x}^{3}+2{x}^{2}
and carry the
−1-1
as well.
Step 4: Divide
−2x2-2{x}^{2}
by
xx
to give
−2x-2x
. Write this next to
x2{x}^{2}
above the line.
Step 5: Multiply
−2x-2x
by
x+2x+2
to give
−2x2+−4x-2{x}^{2}+-4x
. Write this below
−2x2-2{x}^{2}
.
Step 6: Subtract
−2x2+−4x-2{x}^{2}+-4x
from
−2x2-2{x}^{2}
to give
4x4x
. Write this below
−4x-4x
and carry the
−1-1
again.
Step 7: Divide
4x4x
by
xx
to give
44
. Write this next to
−2x-2x
above the line.
Step 8: Multiply
44
by
x+2x+2
to give
4x+84x+8
. Write this below
4x+14x+1
.
Step 9: Subtract
4x+84x+8
from
4x−14x-1
to give a remainder of
−9-9
.
Step 10: Write the final answer:
−x2−2x+4−9x+2-{x}^{2}-2x+4-\frac{9}{x+2}

What's Next

Interested in learning more about polynomial division? Start with our polynomial division problems at the top of this page. Our practice questions let you tackle problems at your own pace. If you get stumped, click on "Solution" to see exactly how we arrived at the answer. Want even more help? Sign up for Cymath Plus today.
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