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.

x² − 12xy + 36y²

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 polynomial we're given is H squared minus 14 H K plus 49 K squared. And our answer choices are answer choice. A, this polynomial is prime answer choice B open parentheses, H minus 14 K closed parentheses squared. Answer choice, C open parentheses, H minus seven K, close parentheses, squared and answer choice. D open parentheses, H minus 13 K, close parentheses squared. The first step in factoring is that we want to look at each term and see if they have any factors in common, if we could factor anything out. And they don't all have a coefficient, we can't factor out of coefficient and they don't all have an H or A K. So there's nothing in common between them that we could factor out. The next step is we will do essentially the reverse of the foil process. We set up two sets of parentheses and then we are going to ask ourselves what times what equals H squared. Those are going to be our firsts. And for H squared, that is going to be a church times. H And now with the first stun, we ask ourselves what times what equals a positive 49 K squared. Those are going to be our last and the positive 49 K squared for the case where that's going to be K times K for 49 that could be a one in a 49 or seven and a seven. But looking at our answer choices, they are looking for factors or sets of parentheses that are multiplied by themselves. So they should be identical. So what are the factors of 49 that are identical? And that would be seven and seven. So we have a seven K and a seven K for our two lasts. And now to determine whether that seven K is positive or negative, we need to look at our middle term for this try no meal, which is negative 14 H K. We get that negative 14 HK when we multiply our outsides, which is the first from the first set of parentheses times the last in the second set of parentheses. And then add that to our insides multiplied together, which is the last from the first set of parentheses. Times are multiplied by the first in the second set of parentheses. Those are our insides. When we add those together, those should equal negative 14 H K and to get them to equal something negative are seven K's are lasts, will need to be negative as well. So now we can foil this to see if we have done it correctly. So our first H times H that's H squared, our outsides H times negative seven K, that's going to be minus seven HK our insides negative seven K times H that's also minus seven HK. And then our last negative seven K times negative seven K, which is going to be a positive 49 K squared, combining our like terms in the middle. Here, we have a negative seven H K minus seven HK and that will combine to equal negative 14 HK and bringing down our firsts that H squared, we have H squared minus 14 HK. Bringing down our lasts plus 49 K squared. These do match. So we have factored correctly and we know that when we have a church minus seven K times are multiplied by H minus seven K that equals parentheses, open parentheses, H minus seven K, closed parentheses squared. So we look at our answer choices for H minus seven K squared and that is going to be answer choice. C 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. x² − 12xy + 36y²

Key Concepts

Perfect Square Trinomials

Factoring Polynomials

Prime Polynomials

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