Perform each indicated operation. See Examples 2 through 8 . (5 x^4+3 x^2-2)/(x-1) ·(x+1)/(x^4-1

Perform each indicated operation. See Examples 2 through 8 ....

Key Concepts

Polynomial Factorization

This concept involves rewriting polynomials as products of simpler polynomials or factors. Techniques such as factoring by grouping, extracting common factors, and using identities like the difference of squares are key to simplifying complex expressions and revealing cancelable factors.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Mastery of operating with rational expressions includes understanding how to multiply, divide, add, and subtract them, while keeping in mind the restrictions on the variable values that would make any denominators zero.

Multiplication of Rational Expressions

Multiplying rational expressions involves multiplying their numerators together and their denominators together. Once the multiplication is done, further simplification typically requires factoring the resulting polynomials to cancel out any common factors between the numerator and denominator.

Canceling Common Factors

After factoring both the numerator and the denominator, any common factors found in both can be canceled. It is critical to note that cancellation is valid as long as the factors being canceled are not equal to zero, which maintains the integrity of the original expression's domain.

Domain Restrictions

This concept pertains to identifying values for the variable that make any denominator zero, thus rendering the expression undefined. When simplifying expressions, it is important to maintain these restrictions even if the factors causing them cancel out.

Transcript

00:01 Okay, so here we have, well, 8 over x squared plus 6x plus 5, and then we are subtracting, so minus 3x over, well, x squared, plus 4x, minus 5, and then we are adding, so plus 2 over x squared, minus 1.

00:34 Okay, so basically adding three fractions.

00:38 I mean, 8 minus, you thought it as 8 plus a minus a negative 3x over this.

00:44 So we're adding three fractions, so therefore we need to have a common denominator.

00:50 So to do so, what we want to do is want to, well, factor.

00:54 I mean, these denominators are all pretty much crying out to be factored.

00:57 So we factor that denominator is what we get here is, is what we still have an 8 on top here.

01:04 And then we factor the denominator here that can factor as well let's see x squared is x and x 5 is well 5 times 1 and then i have a plus 6x in the middle so these are both positive that will give me a 6x so i have x squared plus 6x plus 5 factors as x plus 5 times x plus 1 okay, so i have 8 over x plus 5 times x plus 1 and then subtracting 3x over, well, this denominator can factor as well as, well, x squared is an x and x.

01:46 5 again is 5 times 1, but here i need different signs to get to a negative 5.

01:51 I have to add to a positive 4 in the middle, so i need my positive on my 5 and my negative on my 1.

01:56 All right? so there is my second denominator factored, and then i've got, well, plus 2 over, well, x squared minus 1, that's the difference of all perfect squares that factors as x plus one times x minus one.

02:15 Okay, now what is our common denominator? well, what are my factors in my denominator? i have an x plus five.

02:24 I have an x plus one and i have an x minus one, right? so my common denominator is x plus five times x plus one times x minus one to get that common denominator for my first fraction, i already have an x plus five.

02:39 I already have an x plus one.

02:40 I just need a factor of x minus one.

02:43 So i multiply my first fraction by, well, just by one, so i don't change it, but by one in the form of x minus one over x minus one.

02:52 So then give me an eight, an eight times x minus one on my top, and that's going to be, well, over the common denominator.

03:03 So i can just leave this, because this is all going to be over the common denominator of x plus 5 times, well, x plus 1 times x minus 1.

03:11 So i have 8 times x minus 1, and then i have minus well 3x times here i already have in factor of x plus 5, a factor of x minus 1.

03:22 I need a factor of x plus 1.

03:23 So i have minus negative 3x times well x plus 1 because i'm multiplying this fraction by x plus 1 over x plus 1.

03:36 So then i'll have x plus 5 times x minus 1 times x plus 1, which is what i want as my common denominator.

03:43 And then i have a plus well 2.

03:46 And in this one, while i need a factor of x plus 5, i already have a factor of x plus 1 and x minus 1.

03:52 So i have 2 times or the factor of x plus 5.

03:58 Okay.

03:59 And then this is all over my common denominator of, well, you can think of it as x plus one times x minus one times x plus five okay so i mean this really i mean this could be um i guess it could be a final answer um but i would definitely probably want to clean this numerator up you could definitely distribute this out so if we clean up the numerator what we get at eight distributes and we get an eight x 8x minus 8 and then we get well minus 3x squared so minus 3x squared um right and then we get minus 3x times minus 1 that's minus 3x so minus 3x and then i get a plus 2x that 2x that 2 distributes a plus 2x and then 2 times 5 is a plus 10 so plus 10 okay so there's my numerator um if i clean that up well, i probably want to put this in decreasing power.

05:11 So i can put the negative 3x squared out front...

Please give Ace some feedback