Multiply or divide as indicated. Simplify all answers. (2...

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, adding, subtracting, multiplying, and dividing, is fundamental for solving equations and calculus problems that involve polynomial relationships.

Division of Rational Expressions

Dividing rational expressions involves converting the division into multiplication by taking the reciprocal of the divisor. This procedure requires careful manipulation of the expressions, ensuring that the conversion respects the properties of fractions and that the domain restrictions are maintained.

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of simpler polynomials. This technique is crucial for simplifying rational expressions as it enables the identification and cancellation of common factors between the numerator and the denominator, paving the way for further reduction of the expression.

Simplifying by Canceling Common Factors

After factoring, any common factors that appear in both the numerator and the denominator can be canceled, which simplifies the rational expression. This step is important to reduce expressions to their simplest form, making them easier to analyze or use in further mathematical operations.

Transcript

00:01 For problem number 18, we are asked to simplify the expression with multiplication and also to list the numbers in our domain that have to be excluded.

00:11 Because we have an rational expression, we want to make sure that we do not have zero in our denominator.

00:19 So we have x squared minus 4 over x squared minus 4x plus 4 times 2x.

00:30 Minus 4 all over x plus 2.

00:34 Now because your textbook does want you to include the excluded domain numbers for your original problem without you multiplying and simplifying, all we're going to do is take our two denominators and set those equal to zero first.

00:49 So x squared minus 4x plus 4 equals 0 and x plus 2 equals 0.

00:56 Now i'm not going to do anything for this one.

00:59 I'm just going to write out our factors for x squared minus 4 plus 4, you can go back and look at previous problems in this section that i've completed, and you can see the tricks that i use for these.

01:09 So for this one, this would be x minus 2 times x minus 2 equals 0.

01:16 And because this one's just x plus 2 equals 0, we can go ahead and solve x equals negative 2.

01:21 Now because both of these are x minus 2 equals 0, we know that they're both going to be x equals 2, so there's no need for us to write it twice.

01:30 So our answer for the excluded numbers in our domain would be x is not equal to negative 2 or positive 2.

01:40 And that's the first part to our answer...

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