Simplify. (2 b^3+b^2-2 b+3)(b+1)^-1

Name: Problem 26, Chapter 6: Algebra 2
Uploaded: 2020-03-26T20:08:14-08:00
Duration: 6 min 20 s
Channel: Megan Ludwinski
Description: Simplify. (2 b^3+b^2-2 b+3)(b+1)^-1

Key Concepts

Negative Exponents

A negative exponent indicates that the base is taken as the reciprocal. This means that an expression like (b+1)?¹ is equivalent to 1/(b+1), a key step in rewriting and simplifying expressions involving division.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying such expressions often involves rewriting them, factoring, and canceling common factors to express them in their simplest form.

Polynomial Division

Polynomial division, which can be performed by long division or synthetic division, is a method used to divide one polynomial by another. This concept is crucial when simplifying rational expressions, as it helps to separate the expression into a polynomial part and a remainder, making the overall expression easier to work with.

Transcript

00:01 We are simplifying the expression 2b cubed plus b squared minus 2b plus 3 times b plus 1 to the negative 1 power.

00:11 I'm going to rewrite that second factor where we have that negative 1 so we can use the quotient property, which just says that we can get rid of that negative in the exponent by flipping or finding the reciprocal and changing the sign of our exponent.

00:32 So for this first part, in general, x to the negative a, we can really rewrite as a fraction over one.

00:46 So when we flip our fraction, that one in the denominator now becomes a one in the numerator, and the negative, the x to the negative a power goes from the numerator to the denominator to the denominator, and you'll notice we have a negative a and it turns into a positive a.

01:03 So really, we can just flip the sign there.

01:07 And when a equals 1, we can still use that same property.

01:15 When we flip, that 1 goes from the denominator to the numerator.

01:19 And on our new denominator, we don't have to rewrite that 1 because anything to the first power is just itself.

01:29 So we're going to be rewriting b plus 1 to the negative first power.

01:39 And we can use this part of our property, the x to the negative one.

01:47 So again, anything can be rewritten as a fraction over one.

01:52 So this is a fraction over one.

01:54 So when we flip it, that one becomes our numerator, and the b plus one is our denominator, and we do not have to rewrite that negative one, because now it's positive when we flip it, and anything to the first power is just itself.

02:09 So we're going to replace.

02:13 So this is equivalent to 2b cubed plus b squared of 3 minus 2b plus 3 times 1 over b plus 1.

02:33 So when we combine those two, this is really a fraction over 1.

02:38 So we can multiply the numerators, multiply the denominators, anything times 1 is just itself.

02:44 So we really are left with a fraction.

02:47 2b cubed plus b squared minus 2b plus 3 all over b plus 1.

03:04 So fraction indicates division.

03:07 So we're going to be using division.

03:09 I'm going to use synthetic division.

03:11 So to set that up, we're going to be dividing by b plus 1.

03:15 We take the added inverse.

03:17 So inverse means opposite.