Simplify. (a^3-2 a^2+2 a-4)/(a^3-2 a^2-3 a+6)

Name: Problem 98, Chapter 6: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-06-27T17:55:35-08:00
Duration: 3 min 45 s
Channel: Christine Tabor
Description: Simplify. (a^3-2 a^2+2 a-4)/(a^3-2 a^2-3 a+6)

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Key Concepts

Polynomial Division

Polynomial division, including long division or synthetic division, is a method used to divide one polynomial by another. It can be particularly useful when determining factors or verifying results obtained by factorization, thus playing a key role in simplifying complex rational expressions.

Simplifying Rational Expressions

This concept involves reducing a fraction that contains polynomials in the numerator and denominator by factoring both parts and canceling any common non-zero factors. It is important to consider domain restrictions while canceling to ensure the simplification is valid.

Polynomial Factorization

This is the process of rewriting a polynomial as a product of simpler polynomials. It involves techniques such as grouping, synthetic division, and recognizing patterns like the difference of squares or sum/difference of cubes, which are essential when simplifying rational expressions.

Factor Theorem

The factor theorem states that if a polynomial f(x) equals zero for some value r, then (x - r) is a factor of the polynomial. This principle is used to identify possible factors in both the numerator and denominator of a rational expression, allowing for the cancellation of common factors.

Transcript

00:01 All right, so we need to simplify this.

00:02 And it looks very complicated, but we can do it.

00:06 So i love these with the a cubed and there's an a squared and there's an a and there's a constant because i feel like it's a puzzle.

00:16 And so to solve for these, because there are even terms, i'm going to group these together.

00:25 I'm going to group these together.

00:28 Do the same with the denominator.

00:31 Okay, i'm grouping them in hopes that it turns out.

00:37 So in the first group, a -cube minus 2a -squared, what do they both have in common? they have an a -squared.

00:49 A -cubed, right, is a -times a times a times a.

00:54 If i take a squared out, i'm left with just a.

00:57 Subtract well two times a squared if i take the a squared out i have two left over plus now what can i factor with this second group well they both have a two in common two times a if i factor out i get an a subtract right four is two times two factor out a two i'm left with a look at this beautiful thing.

01:25 In both of those, my parentheses is the same thing.

01:29 A subtract 2.

01:31 That's what you want.

01:33 Because now what's outside the parentheses can be grouped together and what's inside the parentheses can be grouped together or combined together.

01:45 So that is our numerator...

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