Simplify the rational expression. (y^2-3 y-18)/(2 y^2+7 y+3)...

Name: Problem 22, Chapter 0: Algebra and Trigonometry
Uploaded: 2019-06-26T23:10:54-08:00
Duration: 3 min 24 s
Channel: Taylor Shimono
Description: Simplify the rational expression. (y^2-3 y-18)/(2 y^2+7 y+3)

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Key Concepts

Rational Expression

A rational expression is a fraction where both the numerator and denominator are polynomials. Simplifying such expressions usually involves factoring the polynomials and reducing any common factors while being mindful of the restrictions on the variable that could cause the denominator to be zero.

Factoring Quadratic Polynomials

Factoring quadratic polynomials involves rewriting them as a product of two binomials, if possible. This is a key step in simplifying expressions because it allows one to identify and cancel common factors in the numerator and denominator.

Cancellation of Common Factors

Once the polynomials in the numerator and denominator have been factored, any factors that appear in both can be canceled, provided they are not zero. This cancellation simplifies the expression and leads to a reduced form.

Domain Restrictions

When simplifying rational expressions, it is important to consider domain restrictions. These restrictions ensure that the values of the variable do not make the denominator equal to zero, as division by zero is undefined.

Transcript

00:01 So for this problem, we are going to be simplifying this expression.

00:05 And to do so, we are going to want to factor both the numerator and the denominator.

00:10 So i'll go ahead and factor the numerator in red using our diamond and rectangle method.

00:15 We know that the top number in our diamond will be the coefficient of the y squared term and the final term with no y on it.

00:24 So that'll be negative 18 times 1, which is negative 18.

00:29 And the bottom term in our rectangle is going to be given to us by the coefficient of the y term, which is negative 3.

00:38 And so we know that to fill in the rest of our diamond, we want to find two numbers that are going to add to be negative 18, or multiply to be negative 18, and add to be negative 3.

00:50 So those two numbers are negative 6 and positive 3.

00:54 And to finish factoring this, we are going to go ahead and use our.

00:59 Rectangle method.

01:01 So our top left box is going to be the first term in our original expression, which is y squared.

01:09 And the bottom right box is going to be the last term in our original expression, which is negative 18.

01:15 And the other two boxes will be the negative 6 and positive 3 from our diamond and multiplied by a y variable.

01:25 So negative 6y and positive 3y...

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