Perform each indicated operation. Simplify if possible. See Examples I through 7. (x)/(x^2-1)-(2

Perform each indicated operation. Simplify if possible. See...

Key Concepts

Polynomial Factoring

This concept involves rewriting a polynomial as a product of simpler expressions or binomials. It is essential in simplifying rational expressions, as identifying factors such as the difference of squares or perfect square trinomials allows for the cancellation of common factors and makes combining expressions more straightforward.

Least Common Denominator (LCD)

The LCD is the smallest algebraic expression that contains each factor found in the denominators of the given rational expressions. Finding the LCD is crucial when performing operations like addition or subtraction on rational expressions because it allows you to write each term with a common denominator, thereby facilitating their combination.

Simplification of Rational Expressions

This process involves reducing a complex rational expression to its simplest form by combining like terms, cancelling common factors from the numerator and denominator, and ensuring that any restrictions on the variable (such as values that would make the denominator zero) are noted. It is a key algebraic skill for efficiently handling and understanding expressions in algebra.

Transcript

00:02 All right, so we're going to perform this operation here, and it is to subtract these two fractions.

00:09 And we have to make sure that they have the same denominator, so common denominator.

00:17 And these are not written in factored form.

00:20 So i'm going to factor these two, so i can figure out what each denominator is missing.

00:25 So this is a difference of squares here.

00:29 So to fact a difference of squares, i take the square root of the first.

00:33 Term plus the square root of the second term multiplied by the square root of the first term minus the square root of the second term and that's what ends up giving me my difference of squares and then here with my second factor or second fraction i am going to have a factors it's going to be two subtractions because that's going to give me a negative middle term but a positive c term so this is going to be x and x and then one and one so the good news is they both have an x minus one but this one has an x plus one and this one has another x minus one so all three of those colors have to be in both denominators so the first fraction is missing that green and the second factor fraction is missing that blue so i'm going to create equivalent fractions or you have a common denominator here.

01:38 So i'm going to take x over x squared minus one.

01:44 Actually, i'm going to write it out in factored form, just to make it a little easier to see.

01:52 X plus one, x minus one.

01:56 I'm going to multiply by the form of one, by one in the form of the missing factor, which is that green.

02:04 So x minus one over x minus one.

02:08 And that's going to give me my numerator x times the quantity x minus one over my common denominator which is going to be x plus one x minus one x minus one and now i'm going to do that with my second fraction uh so i get two over x minus one times x minus one i'm going to multiply by one in the form of my missing factor, which is that blue factor of x plus one.

02:50 And then my numerator becomes two times the quantity x plus one.

02:56 And my denominator becomes the common denominator of x plus one.

03:01 That looks like an h.

03:04 There we go.

03:05 And then x minus one times x minus one.

03:09 So now i'm going to rewrite this using the fractions with the common denominator.

03:18 So i get x minus one.

03:20 So i get x times x minus 1 over i'm going to write the x minus 1 is x minus 1 squared so i'm going to do x plus 1 and then x minus 1 squared and then i'm going to subtract that second fraction to x minus 1 over x plus 1 x minus 1 squared now something else i'm going to do is i am going to rewrite this subtraction problem as an addition problem.

04:00 And what this is really saying is that i'm subtracting the entire fraction...

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