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

Key Concepts

Polynomial Factorization

This is the process of rewriting a polynomial as a product of its factors. Factoring is critical for simplifying rational expressions, as it allows one to identify common factors that can be canceled. It involves techniques such as factoring quadratics, grouping, or using the quadratic formula when necessary.

Division of Rational Expressions

Dividing rational expressions involves multiplying by the reciprocal of the divisor. This step transforms a division problem into a multiplication problem, after which the expressions can be simplified by canceling common factors before performing the multiplication.

Cancellation of Common Factors

Once the expressions have been factored, any factor that appears both in the numerator and the denominator of the combined rational expression can be canceled. However, it is important to remember that cancellation is only valid when the canceled factors are non-zero.

Domain Restrictions

Throughout the process of simplifying rational expressions, it is crucial to consider the domain of the variables involved. Domain restrictions ensure that any values which cause a denominator to be zero are excluded, as division by zero is undefined.

Transcript

00:01 Before we can add these together, we're going to need to factor, figure out what's missing for the common denominator.

00:07 So the first thing we have here is the r squared plus 7r minus 30.

00:10 So we have to think of things that are going to multiply to 30 that are going to add to 7, and that would be 10 and 3.

00:16 So we're going to do r plus 10 and r minus 3.

00:19 On the right, when multiplies the 10, that adds to 11 is r plus 10 and r plus 1.

00:24 So what that tells me is the left side here.

00:26 I'm going to have to do r plus 1 in the numerator and in the denominator.

00:30 And on the right hand side we're going to have to do r minus three so r minus three in the numerator and r minus three in the denominator so for this fraction we're going to put all the pieces here so r plus one r plus 10 and r minus three and the numerator i'm actually going to go ahead and foil so r times two r is two r squared outside gets you negative 5r inside gets you positive two r so we have negative three r and one times negative five is negative five plus so again we're got the same denominator plus 1, r plus 10, and r minus 3.

01:04 Again, we're a foil.

01:05 So 8 r times r is 8 r squared.

01:08 Outside is negative 24, inside is negative 3.

01:11 Looks like we got ourselves negative 27r, and then negative 3 times negative 3 is positive 9.

01:17 Now we're ready to put these together.

01:19 So when i put them together, again, i'm going to leave in factored form, so r plus 1, r plus 10, and r minus 3.

01:26 So now i'm going to add...

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