Simplify. (b^4-2 b^3+b^2-3 b+4)(b-2)^-1

Name: Problem 9, Chapter 6: Algebra 2
Uploaded: 2020-03-02T05:31:00-08:00
Duration: 3 min 24 s
Channel: Brian Schumeyer
Description: Simplify. (b^4-2 b^3+b^2-3 b+4)(b-2)^-1

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, typically of lower degree, resulting in a quotient and possibly a remainder. This concept is analogous to long division with numbers and is key to simplifying expressions where a polynomial is divided by a binomial.

Multiplicative Inverse

The concept of a multiplicative inverse in algebra refers to a value that, when multiplied by the original number or expression, yields one. In the given problem, (b-2)?¹ represents the reciprocal of (b-2) and is used to transform the division into a multiplication, thereby facilitating the simplification of the expression.

Rational Expressions

A rational expression is a quotient of two polynomials. Simplifying a rational expression typically involves dividing the numerator by the denominator, factoring both parts when possible, and canceling any common factors. This process is fundamental in algebra for reducing expressions to their simplest form.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a linear factor and provides an efficient alternative to long division when the divisor is of the form (b - c). It simplifies the process by focusing on the coefficients, making it quicker and less error-prone when simplifying rational expressions.

Transcript

00:01 Okay, we're going to solve this polynomial b to the fourth minus 2b cubed plus b squared minus 3b plus 4 multiplied by b minus 2 to the negative first.

00:13 So if you think about it, this is really not multiplication because b to the negative, b minus 2 to the negative first is actually 1 over b minus 2.

00:26 So this is a division problem.

00:29 We can do one of two things.

00:30 We can do some trial and error doing some synthetic division, but if b minus 2 is not actually a factor of this, we're going to be here for a very long time trying to figure that out.

00:40 So we could just divide using long division.

00:45 So i could set this up as b minus 2 divided into b to the fourth minus 2 b squared minus 3b plus 4.

01:03 And the way that we divide polynomials using long division is very similar to just doing regular long division.

01:11 We're gonna divide, we multiply, we subtract, and we start bringing down.

01:15 So you're only gonna divide this b into b to the fourth.

01:20 And it goes in b cubed times.

01:22 When you do your multiplication, if you get a remainder here, something went wrong, but you also have to remember you're distributing at the same time.

01:31 B cubed times negative 2 is negative 2b cubed.

01:38 And then subtract.

01:39 Coincidentally, we don't have a remainder here.

01:42 So we can actually bring this down now and do the exact same step.

01:52 B can go into b squared, b times, and distribute...

Please give Ace some feedback