Perform the addition or subtraction and simplify. (1)/(x+1)-(2)/((x+1)^2)+(3)/(x^2-1)

Name: Problem 58, Chapter 0: Algebra and Trigonometry
Uploaded: 2019-06-27T17:02:16-08:00
Duration: 3 min 45 s
Channel: Taylor Shimono
Description: Perform the addition or subtraction and simplify. (1)/(x+1)-(2)/((x+1)^2)+(3)/(x^2-1)

Perform the addition or subtraction and simplify....

Key Concepts

Common Denominator

When adding or subtracting rational expressions, the identification and use of a common denominator is crucial. This involves rewriting each term so they all share the same denominator, allowing the numerators to be combined into a single expression. This is often facilitated by factoring denominators to reveal their underlying structure.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying such expressions often involves operations such as factoring, finding common denominators, and performing arithmetic operations like addition or subtraction on the fractions.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into products of simpler polynomials that can reveal hidden structures, such as identifying common factors. In many cases, recognizing special products like the difference of squares is essential for simplifying rational expressions.

Transcript

00:01 So to start this problem, we are going to first need to make sure that all three of our fractions have the same common denominator.

00:08 So we can go ahead and do that using our method of finding the least common multiple.

00:16 So our first fraction has x plus 1 in the denominator.

00:20 Our second fraction has two x plus 1s.

00:24 And our third denominator, we can factor into, i'll put an arrow, x plus 1 times x minus 1 and so when we look at all of these three we can see that we need two x plus 1 terms and 1 x minus 1 term so the least common multiple that's going to become our common denominator is going to be x plus 1 squared times x minus 1 so in order to get to that common denominator we are going to have to take our first fraction our 1 over 1 over x plus 1, and multiply that by x plus 1 times x minus 1 over x plus 1 times x minus 1.

01:13 And then next we have our second fraction, which is 2 over x plus 1 squared.

01:19 So we know that we are only missing this x minus 1 term.

01:23 So we're going to multiply by x minus 1 over x minus 1.

01:28 And finally, our third fraction is 3 over 3 over, we said it factors into x plus 1 times x minus 1.

01:37 And all we're missing is one more x plus 1.

01:41 So we are going to multiply by x plus 1 over x plus 1.

01:47 And our next step is going to be foiling out all of these terms and distributing each of them.

01:54 So our first term, we have a 1, so we can essentially just ignore that.

01:59 And when we distribute, we will get x squared.

02:02 This is going to be minus x plus so those will cancel minus 1.

02:08 And our next term we have it's subtracting, but if we want to make it easier on ourselves, we can change this to a plus and make it a negative 2.

02:17 And that's going to make it easier when we are distributing our numbers...

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