Perform each indicated operation. (5 x^2(x-1)-3...

Key Concepts

Polynomial Factoring

This concept involves rewriting a polynomial as a product of its factors, which are simpler polynomials or monomials. Factoring is a critical step when simplifying rational expressions as it often reveals common factors in the numerator and the denominator that can later be cancelled.

Canceling Common Factors

After factoring polynomials, any factors that appear in both the numerator and the denominator of a rational expression can be cancelled out, provided they are not equal to zero. This step simplifies the expression and is fundamental in reducing complex fractions to simpler forms.

Multiplication of Rational Expressions

Multiplying rational expressions involves multiplying their numerators together and their denominators together. It is important to factor each polynomial first, as this may reveal opportunities to cancel common factors before or after the multiplication, leading to a simplified result.

Simplification of Rational Expressions

Simplification is the process of reducing a complex rational expression to its simplest form by factoring, canceling common factors, and performing any necessary operations. The goal is to present the expression in a more concise and easily understandable manner.

Domain Restrictions

When working with rational expressions, it is important to identify values of the variable that make the denominator zero, as these are not permitted. Domain restrictions ensure that the expression remains valid and provide clarity about the range of values for which the simplification holds.

Transcript

00:01 Okay, so for this problem, we have expression 3 over x plus, or sorry, that should be minus, 3 over x minus 2x over x squared minus 1 plus 5 over x plus 1.

00:24 Now, if we take a look at this, we see we want to subtract, do some addition, but we can't because we don't have a common denominator.

00:31 That's what we're missing here.

00:33 So let's try to figure out what the common denominator is and kind of add that in.

00:37 So start off, let's just kind of factor as much as we can.

00:41 So 3 over x, nothing to factor there, minus 2x over, oh, x squared minus 1.

00:49 Looks like we can do some factoring there.

00:51 We're going to have x minus 1 and x plus 1.

00:56 And if we kind of foil that out, we'll see we'll get the x squared minus 1.

00:59 And we know this because the first term of both of these has to be second term has to be one and it has to be opposite signs because we have a minus one.

01:07 Then we have that plus 5 over x plus 1.

01:14 So now we can kind of take a look at what the common denominator might be.

01:23 So we know for sure we're going to have an x because that's what we have here.

01:27 We know we need an x minus 1.

01:29 We know we need an x plus 1 because those two are right here.

01:33 Now we have the x plus 1 over here.

01:34 But we don't really need that in our common denominator just because it's already taken into account.

01:39 So let's see, what do we do to get the common denominator on all of these? here we need to multiply by x minus 1 to the numerator and the denominator.

01:50 So we do that.

01:51 Oh, and it looks like we're also missing the x plus one.

01:56 So we can do that one as well.

02:01 Here, all we're missing is the x.

02:03 So we multiply the numerator and the denominator by x.

02:07 And then here we're missing the x and the x minus 1.

02:10 So x x minus 1, x times x minus 1.

02:18 So yeah, let's multiply all these out and see what we end up with.

02:22 So out here we're going to have three times x minus 1 times x plus 1, all that over, let's just say common denominator for now, so we don't have to write it out too many times...

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