How to divide polynomials

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ACT Math › How to divide polynomials

Questions 1 - 3

1

What is $\frac{x^{2}-x-30}{x^{2}-4x-12}$ equal to?

$\frac{x+5}{x+2}$

$\frac{x-5}{x-2}$

$\frac{x-6}{x+2}$

$\frac{x+5}{x-6}$

Explanation

1. Factor the numerator:

$x^{2}-x-30= (x+5)(x-6)$

2. Factor the denominator:

$x^{2}-4x-12=(x-6)(x+2)$

3. Divide the factored numerator by the factored denominator:

$\frac{(x+5)(x-6)}{(x-6)(x+2)}$

You can cancel out the from both the numerator and the denominator, leaving you with:

$\frac{x+5}{x+2}$

2

Simplify:

$\frac{x^2+8x-9}{x^2-6x+5}$

$\frac{x+9}{x-5}$

$\frac{x+9}{x-1}$

$\frac{x-9}{x-1}$

$\frac{x-1}{x+5}$

$\frac{8x-9}{-6x+5}$

Explanation

In order to divide these polynomials, you need to first factor them.

and

Now, the expression becomes $\frac{(x+9)(x-1)}{(x-5)(x-1)}$

$\frac{x-1}{x-1}=1$

so,

$\frac{(x+9)(x-1)}{(x-5)(x-1)}=\frac{x+9}{x-5}$

3

Simplify the following division of polynomials:

$\frac{x^2+3x+3}{x+1}$

$x+2+\frac{1}{x+1}$

It cannot be simplified any further.

$x+\frac{3}{x+1}$

Explanation

The leading term of the numerator is one exponent higher than the leading term of the denominator. Thus, we know the result of the division is going to be somewhere close to $\frac{x^2}{x}=x$ . We can separate the fraction out like this:

$\frac{x^2+3x+3}{x+1} = \frac{x^2+x}{x+1}+\frac{2x+3}{x+1}$

The first term is easily seen to be $\frac{x(x+1)}{(x+1)}$ , which is equal to . The second term can also be written out as:

$\frac{2x+2}{x+1}+\frac{1}{x+1}=2+\frac{1}{x+1}$

and combining these, we get our final answer,

$x+2+\frac{1}{x+1}$

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