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(x^3-7x^2-7x+20)/(x+4)

2 min read Jun 17, 2024
(x^3-7x^2-7x+20)/(x+4)

Factoring and Simplifying the Expression (x^3 - 7x^2 - 7x + 20) / (x + 4)

This article will guide you through the process of factoring and simplifying the rational expression (x^3 - 7x^2 - 7x + 20) / (x + 4).

1. Factor the Numerator

We can factor the numerator, x^3 - 7x^2 - 7x + 20, using various methods. One common method is polynomial long division or synthetic division.

Using synthetic division:

  • Set up: Write the coefficients of the numerator (1, -7, -7, 20) and the opposite of the constant term in the denominator (-4).
  • Divide: Perform synthetic division:
 -4 | 1  -7  -7  20
      -4  44 -148
     ----------------
       1 -11  37  -128
  • Result: The result shows that the factored form of the numerator is (x + 4)(x^2 - 11x + 37) - 128.

2. Simplify the Expression

Now we have:

(x^3 - 7x^2 - 7x + 20) / (x + 4) = [(x + 4)(x^2 - 11x + 37) - 128] / (x + 4)

Notice that (x + 4) appears in both the numerator and denominator. We can cancel them out, but only if x ≠ -4.

Therefore, the simplified expression is:

**x^2 - 11x + 37 - 128 / (x + 4) **

OR

**x^2 - 11x - 91 / (x + 4) **

Important Note: It is crucial to remember that the simplified expression is valid for all values of x except x = -4. This is because the original expression is undefined for x = -4 due to the denominator becoming zero.

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