Solve each equation in Exercises 96–102 by the method of your cho...

Question

Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18

Accepted Answer

Welcome back. I'm so glad you're here. We are solving for X. And now we've got these four terms here, X cubed plus two on one side of the equation two X squared plus X on the other side of the equation. And when we've got four different terms like that of different degrees. One method that works sometimes is factoring by grouping. So we're going to try that here and we're going to get everything on the same side. We'll take the two X squared and scoot it over to the left and the X and scooted over to the left. So that will give us x cubed minus two, X squared minus X plus two, putting those in descending order of their degrees. And so now we can group them together. We'll group these first two terms here and we and say okay, what have they got in common? That we could factor out. Well they both have an X squared so we can take that X cubed divided by X squared is going to leave us with X for the first term and negative two, X squared divided by X squared, leaves us with negative two for the second term. And now looking at these second two or the second set of two terms. The last the third and the fourth, we can say what have they got in common? Well, we could factor out a negative one and negative X divided by negative one leaves us with X and two divided by negative one leaves us with negative two. And this is great because now these terms here, X squared times x minus two and negative one times x minus to have a factor in common. That we can factor out. They both have an x minus two being multiplied. So we can factor an x minus two in front. And that leaves us with the x squared minus one and that's still equals zero. And this is getting pretty close. We've got two factors here, but the second one is actually a difference of squares. So we can factor that one even further. So we'll keep our x minus two. And then how do we get those squares? The X squared and the negative one? Well X square that's going to be X times X and negative one. That's going to be positive one times negative one. And that is as factored as we can get that equation. So now let's set each factor equal to zero. We've got x minus two equals zero and we've got x plus one equals zero and we've got x minus one equals zero. And we'll get X by itself will add to double sides. So one of our solutions is x equals two, will subtract one from both sides. So one of our solutions is X equals negative one and we'll add one to both sides. So one of our solutions is X equals one. We look at our answer choices and that matches with answer choice. D Well done. We'll catch you on the next one

Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18

Key Concepts

Polynomial Equations

Factoring

The Rational Root Theorem

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