Exercises 107–109 will help you prepare for the material covered ...

Question

Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x^3+3x^2−x−3

Accepted Answer

Welcome back. I'm so glad you're here. We are given this expression X to the third plus four, X squared minus 25 X minus 100. And we are asked to factor completely when we've got four terms like this, we can try factoring by grouping. So we'll group these first two terms together. X to the third plus four X squared. And the second two terms together negative 25 X minus 100. We look at the first two terms and we ask ourselves do they have anything in common that we can factor out? And yes, they both have an X squared in them. So X to the third divided by X square just leaves us with X and four X squared divided by X squared. Leaves us with just plus four and then parentheses. And now we look at the second two terms here, the second group and we've got a negative 25 X minus 100. Anything in common that we can pull out? They both have a negative 25 in them. Negative 25 X divided by negative 25. Leaves us with X negative 100 divided by negative 25 Leaves us with plus four and perfect. Both of these groups here have a factor of X plus four so we can factor X plus four out. We put that as a factor X plus four and it is being multiplied by both X squared and negative 25. Well we've got it partially factored but if you look at this X squared minus 25 that is a difference of squares. So I'm going to bring down this X plus four factor and I'm going to put two sets of parentheses here. And we recall from previous lessons that when we have a difference of squares, we are going to take the square root of both of those terms. So the square root of x squared is going to be X. And that X goes in the first spot in each set of parentheses And then we take that squirt of just the 25 here and the squared of 25 is five. We're going to put a five in the second spot in both of those sets of parentheses. And then by definition for our difference of squares, one will be a plus and one will be a minus. And that is as factored as this expression is going to get. So our final answer X plus four times X plus five times X minus five. We look at our answer choices and that matches with answer choice. D Well done. We'll catch you on the next one.

Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x^3+3x^2−x−3

Key Concepts

Factoring Polynomials

Rational Root Theorem

Synthetic Division

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