In Exercises 23–48, factor completely, or state that the polynomi...

Question

In Exercises 23–48, factor completely, or state that the polynomial is prime.x³ + 3x² - 4x - 12

Accepted Answer

Welcome back. I am so glad you're here. We are asked to completely factories the following polynomial expression. If possible. Otherwise declare if the expression is a prime polynomial, the polynomial we're given is Q to the third power plus four, Q squared minus 25 Q minus 100. Our answer choices are answer choice. A open parentheses, Q minus five, close parentheses multiplied by open parentheses. Q plus five, close parentheses multiplied by open parentheses. Q plus four, close parentheses. Answer choice B open parentheses, Q minus four, close parentheses multiplied by open parentheses. Q minus five, close parentheses squared. Answer choice C open parentheses, Q plus four, close parentheses multiplied by open parentheses. Q minus five, close parentheses squared, an answer choice D this polynomial is prime. All right. Well, the first step in factoring any polynomial Xyz, we take a look at all of our terms and we see if there's anything we can factor out in front. Noting that the first term here is Q to the third power and the last one is negative 100. Those do not share any common factors. So we can't factor anything out of all of them. However, we do notice that we have four terms and we can attempt to factor by grouping, we group the first two terms Q to the third power and four Q squared. And the 3rd and 4th terms together negative 25 Q and negative 100. And so what we do is we just look at our groups and now we ask ourselves, is there anything any common factor that we can factor out between Q two, the third power and four Q squared? The greatest common factor is Q squared. So we'll factor out A Q squared. And that leaves behind open parentheses. Q to the third divided by Q squared, leaves us with one Q or just Q and then positive four Q squared divided by Q squared leaves us with just plus four close parentheses. Now looking at the second group here, we ask ourselves between negative 25 Q and negative 100. What is the greatest common factor? And the greatest common factor there is negative 25. So we factor out a minus 25 that's multiplied by what's left from those 3rd and 4th terms negative 25 Q divided by negative 25 leaves us with Q and negative 100 divided by negative 25 leaves us with plus four, close parentheses. And we notice that both Q squared and negative are being multiplied by Q plus four. So there is a common factor here now that we can pull out, we can pull out a open parentheses. Q plus four, close parentheses. And that's being multiplied by both. So we put new open parentheses, Q squared and the negative 25. So we put Q squared minus 25 close parentheses. And now we've got this partially factored. However, we take a look at the second set of parentheses here, Q squared minus 25. And we notice we have a difference of squares. We recall from previous lessons. The difference of squares is when we would have an A squared. So in a term that's squared minus B squared. So A B term that's squared, and we can factor that we put two new sets of parentheses. And by definition, that will factor to be in the first set of parentheses, A plus B and in the second set of parentheses A minus B. And so we can identify our A and R B. So in Q squared minus 25 are a term is the first one being squared. That's cute in our B term, that's the one that's being subtracted and squared. And so that's 25. So the square root of 25 or what's being squared, there is five. So B is five and now we can fill those in to our factoring sets of parentheses in the first set of parentheses instead of A, we have Q and then plus, and instead of B, we have five and then the second set of parentheses instead of A, we have Q and bring down minus and instead of B, we have five again. So we have factored that. Now we can bring down our first set of parentheses. Open parentheses, Q plus four, close parentheses, multiplied by new, open parentheses. We have Q plus five, close parentheses and that's multiplied by open parentheses. Q minus five, close parentheses. So that's as factored as that one's going to get, we look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 23–48, factor completely, or state that the polynomial is prime.x³ + 3x² - 4x - 12

Key Concepts

Factoring Polynomials

Rational Root Theorem

Synthetic Division

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