In Exercises 49–64, factor any perfect square trinomials, or stat...

Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime.9y² + 6y + 1

Accepted Answer

Welcome back. I am so glad you're here. We are asked to completely factories the following perfect square, try no meal. If possible. Otherwise declare if the expression is a prime polynomial, the expression were given is 25 M squared plus 20 M plus four. Our answer choices are answer choice. A this polynomial is prime answer choice. B open parentheses. Five M plus two, close parentheses multiplied by open parentheses. Five M plus two, close parentheses. Answer choice, C open parentheses. Five M plus one, close parentheses multiplied by open parentheses. Five M plus four, close parentheses, an answer choice. D open parentheses 25 M plus one, close parentheses multiplied by open parentheses. M plus four, close parentheses. The first step in factoring any polynomial is that we're going to look at all of our terms and ask ourselves if there are any common factors that we can factor out and between our coefficients between 25 20 and four. Those don't share any common factors and not all three terms have an M. So we can't factor variable out either. So the next thing we'll do is set up to big sets of parentheses and we'll essentially do the reverse of the foil process. We ask ourselves what multiplied by what will give us 25 M squared. Those will be our firsts. And for that M squared, that's going to be M multiplied by M. But that 25, we've got options for 25, could be one multiplied by 25. It could also be five multiplied by five order doesn't matter for the first. So I'm going to leave it with those two options now because we don't know the first. We can move on and get more clues here. We can ask ourselves what multiplied by what gives us a positive four. Those will be our lasts. And for four, again, we've got options and this time order does matter. So we could have a one multiplied by four. We could have a negative one multiplied by negative four. We could have a four multiplied by one, a negative four multiplied by negative one. We could also have two multiplied by two and negative two, multiplied by -2. So how do we know which ones to plug in? Well, we need to discern which ones when we take our outsides, which is the first from the first set of parentheses multiplied by the last, from the second set of parentheses and add that to our insides, which is the last from the first set of parentheses multiplied by the first, from the second set of parentheses. Then that needs to add up to a positive M And noticing that that positive m the second term in our try no meal is positive. We can eliminate all of the negative options from our factors for four. So we can cross off negative one, multiplied by negative four, cross off negative four, multiplied by negative one and cross off negative two multiplied by negative two. And so now that narrows it down a little bit. And the other thing that can help us narrow it down a little bit is this is multiple choice. Normally, you would just start throwing different ones in there and get an idea. A feel for what's getting us close to that 20 m when we add them together but we can look at our answer choices and just choose from the ones that are given. So the first one here is answer choice B which has five M and five M for our firsts. So we can choose those to start with for our first and then for our lasts, they have plus two and plus two. So we can choose those to test out. So in our first set of parentheses, we have five M plus two and in our second set of parentheses, we also have five M plus two. Now we can foil this out to check first, give us five M multiplied by five M and that gives us a 25 M squared which we knew would work. Now testing the outsides five M multiplied by two, that's plus 10 M and the insides two multiplied by five M. That's again plus 10 M. And then our lasts two, multiplied by two. That's plus four, which we knew that one would work. Now, we'll bring down our 1st 25 M squared. Combine our like terms in the middle here, we have 10 M plus 10 M, which is plus 20 M. And then our last springing down this plus four and this does match, we have factored it correctly. So our answer choice for open parentheses, five M plus two, close parentheses multiplied by open parentheses. Five M plus two, close parentheses is answer choice B well done. We'll catch you on the next one.

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime.9y² + 6y + 1

Key Concepts

Perfect Square Trinomials

Factoring Polynomials

Prime Polynomials

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