In Exercises 17–38, factor each trinomial, or state that the trin...

Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−11x+4

Accepted Answer

Hi there. So for this problem says you wanna factor the given polynomial completely or categorize the polish prime. If it doesn't get fact arised furthermore, I'm gonna go ahead and rewrite the polynomial 1st 17 X squared minus X plus 15. I'm going to create two linear factors of this polynomial do that. You need to know um what are factors of our of our coefficients. So the polynomial in this case is a quadratic. You know, quadratic is defined by the formula X squared plus bx plus C. So this case are a 17, r. c is 15. No, we'll check the factors of our A and R. C. We actually know the factors based on our answer list on the left. But I'm just gonna go ahead and show you the factors. So 17 is our first one. Well the fact is 17, 17 is a prime number. That's just one and 17. Now for 15 We have one and And we also have three and 5. A good way to generate illicit factors is to start at one and see if we can divide out a number until we get to a certain point where we run out factors. We start seeing repeat factors like five. We kept going, I would get five and three which we already have But in this case we have one in 17 for 17, 1 and 15 and three and 5 for 15. So we know there are factors then based on our list When I have 17 and one and 15 and one. So let's make two linear expressions multiplying each other. We have 17 X as our first one and X as our other one. Now we need our X. Because this is quadratic and we have an X squared up there And we need two numbers for our 15. So also there's 15 there and one there. So now the goals in terms of what signs are which. So we'll have to check this in a second. But we're gonna test some signs. Now, what I could do was I first like to check to see what the sign of C is. So if you look over here, see is a Plus. We're gonna have to use the foil method to check our multiplication, you foil, if you recall is just the way we will multiply two linear expressions together to get another expression. So with foil we're not multiplying our numbers That will determine what our sign is. We have a plus on the 15. So we're gonna have either have two positive numbers at the end or two negative numbers at the end. So in other words there's 15 is one that you're gonna be both positive or both negative. We don't know which one exactly just yet. We can find it though by looking at the sign of our B. In this case the B is negative. So if we have a positive C. But a negative B. The signs inside of our linear expressions are most likely going to be negative. So we have 17, X -15 and X -1. I'm gonna go ahead and check these using foil. So foil says that we multiply our first terms, multiply our outer terms multiply our inner terms and then multiply our last terms and add them all together to see what we get. Some of them will supply. The first terms. The first terms are the very first terms in our expression. So in this case that's 17 x. And X. Now I'm gonna add now I'm going to do our outside terms. So these are the terms that are outside of The linear sessions. So in this case we have our 17 x Was on the outside of this one and our -1 which is on the inside. I'm sorry the outside of the other one. Now we multiply our inner expressions. So inner And the numbers are inside each of our questions. So we have negative 15 times X. Finally we have our last so it's our last numbers and our expressions Where we have a 215 and they live one that we can simplify this. We're gonna get 17 x squared here we have negative 17 X times negative one is negative 17 X. Neither 15 times x is -15 x. Probably 90 to 15 times negative one is positive 15. So we're going to simplify this. It's 17 x squared -17 X - is -32 x. And we'll just bring down our 15. So since we did that we had the same polynomial we started with, so our final answer then is 17 X - Times X -1. We have confirmed that is the right answer and that answer is answer d. Okay. I have to help you solve the problem. Thank you for watching. Goodbye.

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