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 - 7x² - x + 7

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 11, Q squared minus Q minus 11. And our answer choices are answer choice. A open parentheses, Q minus one closed parentheses multiplied by open parentheses. Q minus 11. Close parentheses multiplied by open parentheses. Q plus 11, close parentheses. Answer choice. B open parentheses, Q plus one closed parentheses multiplied by open parentheses. Q plus one closed parentheses multiplied by open parentheses. Q minus 11, close parentheses, an answer choice C is open parentheses. Q minus one, close parentheses, multiplied by open parentheses, cubed plus one, closed parentheses multiplied by open parentheses. Q plus 11, close parentheses and answer choice D that the polynomial is prime. All right. Well, when we are factoring, the first thing we're going to ask ourselves is if there are any common factors between all of our terms that we can factor out in front. And this time we see that the first term is Q2, the third power. And the last term is negative 11. Those don't share any common factors. So there's nothing we can pull out of all of them. However, we do see with these four terms, we could attempt to factor by grouping, we can group the first two terms Q to the third power and positive 11 Q squared together. And then in these 3rd and 4th terms negative Q and negative 11. And just between those two, in each group, we ask ourselves, do they share anything in common? And so in the first group, Q to the third power and 11 Q squared, we see that between Q two, the third and Q squared, we could factor out a Q squared. And then what's left from each of those two terms, Q two, the third divided by Q squared leaves us with Q plus 11, Q squared divided by Q squared leaves us with 11. So that's Q squared multiplied by Q-plus 11. Now looking at the second group of our two terms here, the 3rd and 4th terms, we have a negative Q minus 11. Well, the only common factor there is a minus one that we are going to factor out negative Q divided by negative one, leaves us with a positive Q and negative 11 divided by negative one leaves us with a positive 11. So we have reading the whole thing Q squared multiplied by Q plus 11 minus one multiplied by Q plus 11. And look at that they are both being multiplied by the Q squared in the negative one by Q plus 11. So that's a common factor and we can factor it out of those two terms. We can put um open parentheses. Q plus 11, closed parentheses multiplied by a new set of parentheses. And that will have Q squared minus one closed parentheses. Alright, we have got this partially factored. However, looking at the second set of parentheses here where this Q squared minus one is, we recognize a difference of squares. And we recall from previous lessons that when we have a difference of squares that would be having parentheses, A squared minus B squared. That will be two new sets of parentheses in the first set. That's going to be A plus B and in the second set, that's going to be A minus B and what are our A and B in this case, we'll A, that's the first term being squared and here A equals Q and B, that's the second term being squared and the one being subtracted. And in this case B equals positive one. So now we can set up, we can fill in our A's and Arby's for these two sets of parentheses. So we have open parentheses instead of A, we have Q plus instead of B, we have one closed parentheses multiplied by open parentheses instead of A, we have Q and then bring down A minus and instead of B, we have one close parentheses. So we factored that Q squared minus one. So we will have three sets of parentheses. The first is Q plus 11 Multiplied by the next set of parentheses is Q-us one. And the last set of parentheses multiplied by Q -1. So we look at our answer choices for Q plus 11 Multiplied by Q-plus one Multiplied by Q -1. And that's going to match with answer choice. C. Well done. We'll catch you on the next one.

In Exercises 23–48, factor completely, or state that the polynomial is prime. x³ - 7x² - x + 7

TextbookSolutions.keyConceptsTitle

Factoring Polynomials

Rational Root Theorem

Synthetic Division

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