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. x^2−14x+45

Accepted Answer

Hi there. This problem says we factor the given polynomial. Really categorize the pollen will be prime if it doesn't get factories further work. So go ahead and rewrite the polynomial. You have X squared minus eight X plus 15. You know the basic form of a quadratic? Which is what this equation is, which is a X squared plus bx plus C. So I'm gonna go about this. So I'm just trying to find factors of each of our coefficients and actually we see the final factors of A. And C. So if you look at this, R. A is equal to one. R. C. Is equal to 15. So I'm going to multiply this A. N. C. Together and figure out what factors are between those two numbers. So I have a as one CS 151 times 15 equals 15. So I'm gonna find the factors of 15. So think of 15. Well we can try one times 15. So I write that down one times 15 equal to 15. Now let's check to see if there's any more factors. Well what I like to do is I like to go down the list starting from one and keep going until I see repeat factors. So 12, 15 is one. I'd go to to next. Well too is not evenly divisible into 15. So I'm gonna skip to let's try three. 3. 3 is evenly divisible 15 which is So we have another factor. Let's check for next or is not evenly divisible c. 15 divided by four. So we go it's five. Well now we have five and three again. so we have a repeat factor so we can stop right there. Now. Now that we have our factors wanna see if either of those added together will equal to RB in the middle. So you want to see if any of these factors will add up to equal R. 8? Well if you look We have 1215, we add those together, we get 16 which isn't going to work but we have three and 53 and five added together will equal eight. So I'm gonna go back to my equation the X square. Now this is actually a -8. So since it is negative, we're actually gonna turn everything negative. So let's say we have a negative three X and a negative five X. And we're gonna add 15 at the end. Okay? So, you know, there were factors are three and 5. And so I'm gonna see if we can factor anything out of these numbers. So we're not gonna do what's called factor by grouping. So I'm gonna go ahead and group these first two terms together and these last two terms together and see if we can factor out anything comin from either factor. So let's look at the first one, you have X squared minus three X. We factor out anything from that question and I say we can if you look closely here, both of these terms have an X. I can actually take out an X. Here, I do that X. We're taking on X from every term X squared taking out X is just X and this is just three. Now we'll look at the other term. So we have negative five x plus 15. Can we take out any numbers out of this? And I say yes we can. Again. I'm actually gonna take out a negative five here. Now how factor by grouping works is when we factor out our numbers we should get some number Like X and -5. We should get those numbers in front of our princes and we should get the same number inside the parentheses or same expression inside the parentheses. So I have to find out of this. We have an X lift here. Alright, this in red. We have an X lift here, factor a negative five from 15. That is -3. So Are two factors here, X -3. X -3. They're the same. Which means we can go ahead and say that this worked Now we're left with two factors X - is one of them and then whatever is remaining on the outside will be combined as one. Factor X minus five, which means that our answer is going to be x times three. I'm sorry X minus three times X minus five which means our answer is answers. See, okay, I want to help you solve the problem. Thank you for watching. Goodbye

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