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.

9x² + 48xy + 64y²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to completely factories the following perfect square trin Amiel. If possible. Otherwise declare if the expression is a prime polynomial, the expression were given is 16 M squared plus 56 M N plus 49 N squared. And our answer choices are answer choice. A, this polynomial is prime answer choice B open parentheses, eight M plus 14 N closed parentheses, squared. Answer choice, C open parentheses, eight M plus seven N closed parentheses squared. Answer choice D open parentheses, four M plus seven N close parentheses squared. And in fact, during the first thing we do is look at all of our terms and see if they have any common factors that we could factor out in front and between our coefficients 16 and 49 those do not share any common factors. And as far as the variables, not all three of them have an M term and not all three of them have an N term. So we can't factor out any variables either. So the next thing we do for factoring is we are going to set up two sets of parentheses and essentially do the opposite are the reverse of the foil process. So we're going to ask ourselves what times what equals 16 M squared. Those will be our firsts. And as far as that M squared is concerned, that's going to be M times M For 16, we have a lot of factors that multiply to equal 16. However, instead of listing them and trying different ones, we notice from our answer choices that they have chosen factors. So sets of parentheses that are identical to each other that are squared. And so what we want to look for is factors of 16 that are identical to themselves and what times what equals 16, that is itself, that's 44 times four equals 16. And that would be the same in both sets of parentheses. So we have, for our firsts, we have four M. Now we turn our attention to our lasts. We ask ourselves what times what equals a positive 49 squared. And for this end squared, that's going to be N times N For the 49. Well, we could have one times 49 or seven times seven. But again, we want something that is multiplied by itself and that is going to be seven times seven. Now, looking at our middle term, that's going to tell us whether those seven ends will be positive or negative. And so they have to take our outsides, which is the first from the first set of parentheses times the last from the second set of parentheses and add those to our insides, which is the last from the first set of parentheses times the first, from the second set of parentheses. And then we add those together and they need to equal a positive 56 M N. So because they're going to multiply to be something positive R seven ends will also be positive. And now we can go ahead and do the foil process to see if this does in fact equal our original perfect square, try no meal. So for our first, we have four M times four M which is 16 M squared by design. Now, for our outsides, we have four M times seven N that's plus 28 M N, our insides, we have seven N times four M that's plus M N. And for our lasts, we have seven N times seven N which is plus 49 N squared. Now combining our like terms here in the middle, we can bring down the 16 M squared and combining those like terms, we have positive 28 M N plus 28 M N which is plus 56 M N And bring down our last this plus and squared and this does match the original perfect square, try no meal. So we have factored this correctly. And now we know we can take four M plus seven N times itself and rewrite that as open parentheses, four M plus seven N close parentheses squared. Looking at our answer choices, this matches with answer choice. D 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. 9x² + 48xy + 64y²

Key Concepts

Perfect Square Trinomials

Factoring Polynomials

Prime Polynomials

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