In Exercises 15–32, multiply or divide as indicated. (x^3−25x)/4x...

Question

In Exercises 15–32, multiply or divide as indicated. (x^3−25x)/4x^2 ⋅ (2x^2−2)/(x^2−6x+5) ÷ (x^2+5x)/(7x+7)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this beautifully long expression here and we get to do the indicated operation and then simplify. So on the first one we've got X squared minus over X plus one squared multiplication is the operation here times X squared plus eight X plus seven divided by two X squared minus. Don't forget. That's a minus there. 15 X plus seven. And now our new operation is division divided by X plus seven squared divided by two X squared plus x minus one. Awesome. So first let's take a look at this division. When we're dividing fractions, we know we're going to flip the numerator and the denominator of the fraction immediately after the division symbol and then that's going to become multiplication. So let's re copy down all this other stuff, X squared minus 49 divided by X plus one squared times X squared plus eight X plus seven divided by two X squared minus 15 X plus seven. And now we flip this one over numerator is now two X squared plus X minus one. And the denominator is now X plus seven squared. Now it's all straight up multiplication. And this is fun because when we are multiplying we want to get everything as factored as possible and then we get to do some beautiful, canceling out of factors. Maybe we'll see. We'll see if we get to. So this first one here, X squared minus 48 49. Excuse me. We recall from previous lessons that that is a difference of squares. So the square. The first one is X. So that X times X. And the second one is 49? Well that's a plus seven and a minus seven. Great that one's done. And now in that denominator we've got X plus one squared. So we can just rewrite that as X plus one times X plus one. Great. Got that one done. Now let's take a look at this X squared plus eight. X plus seven X. Factor that one. So X squared. What multiplies to get X squared? Will that be an X times an X. What multiplies to get plus seven and adds up to B plus eight? Well that would be a plus seven and a plus one. That one's factored. Now we take a look in the denominator two X squared minus 15 X plus seven. What times what gives us two X squared? Will that be a two x times an X? What times what gives us seven? But adds up to be a negative 15. Well that's going to have to be a negative times a negative to get us a positive seven but add up to be negative 15. Wanna get all the way down to negative 15. So that'd be two X times negative seven and then a negative one times X will add up to negative 15 and multiply to get a positive seven, awesome. That one is factored. Now let's take a look at this two X squared plus X minus one. What times what gives us two X squared? Will that be a two X. And an X. What times what gives us negative one? But adds up to be a positive one. Well let's get this positive one here. So two X times one will be a positive two X. And then a minus one. X. Will give us our plus X. For the adding and a negative one for the multiplication. Great. That one is factored. And now the denominator that one's not too tough. We've got an X plus seven times and X plus seven which is X plus seven squared. Cool. And now this super duper fun part. We look at the beginning of our first numerator we have an X plus seven. Are there any X plus plus seven's anywhere in the denominators? Yes there are, they cancel out. And now those are little ones. We have an X minus seven in the numerator. X minus seven. In the denominator yep boom canceled ones X plus seven X plus seven ones. X plus one X plus one ones, two x minus +12 x minus one ones and an X plus one and an X plus one ones. They all became ones that means we've got one times one times one times one times one times one times I lost track. Anyway the top it's all ones the bottom it's all ones, one divided by one is one. We have got answer choice D. one. Well done. We'll catch you on the next one.

In Exercises 15–32, multiply or divide as indicated. (x^3−25x)/4x^2 ⋅ (2x^2−2)/(x^2−6x+5) ÷ (x^2+5x)/(7x+7)

Key Concepts

Polynomial Operations

Factoring Polynomials

Rational Expressions

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