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.

64y² − 16y + 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 121 Z squared plus 66 Z plus nine. And our answer choices are answer choice. A, this polynomial is prime answer choice B open parentheses. 11 Z plus nine, close parentheses squared. Answer choice, C open parentheses. Nine Z plus 11, close parentheses squared. Answer choice, D open parentheses, 11 Z plus three, close parentheses squared. So the first step in factoring is we take a look at all of our terms and ask ourselves if there's anything we can factor out in front and with our coefficients 121 66 and nine, there aren't any common factors between those and not all three terms have a Z. So we can't factor out a variable either. So the next thing we'll do is essentially the reverse of the foil process will set up two sets of parentheses. And we will ask ourselves what multiplied by what equals 121 Z squared. Those will be our firsts and for the Z square, that's going to be easy, multiplied by Z. But what about 121? We've got options. We have one times 121 or 11 times 11. But looking at our answer choices, these are all factors that are squared. So our two sets of parentheses need to be identical to each other. So what factors of 121 are identical to each other? Well, that's 11 and 11. So our firsts are both 11 Z, we've got our first done. So now we ask ourselves what multiplied by what equals the third part of our training meal. This plus nine, those are our lasts. And for nine, again, we've got options one times 93 times three. But the factors that are identical to each other are three and three. The next question is, are those threes positive or negative? Well, we discern that by taking our outsides. So that's going to be this first from the first set of parentheses, multiplied by the last, from the second set of parentheses and add those two, the insides, which is the last from the first set of parentheses, multiplied by the first from the second set of parentheses. And those need to add up to equal this positive Z, the middle term in our Trina meal. And because they add up to a positive 66 Z, we know that these threes can be positive. So our two sets of parentheses both contain 11 Z plus three. Now, we can foil this out to make sure they do. In fact, add up to 66 Z when we multiply the outsides and insides and add them together. So let's start with the 1st 11 Z multiplied by 11 Z which is going to be 121 Z squared. We have our outsides 11 Z multiplied by three, which is plus 33 Z, our insides three multiplied by 11 Z which is plus 33 Z and R lasts three, multiplied by three, which is plus nine. Now, adding our outsides and insides, these like terms will bring down the first, which is 121 Z squared. Adding our outsides and insides positive 33 Z plus 33 Z is plus 66 Z and then our lasts bringing down plus nine and this does match what we started with. So we factored correctly. Now, we know we can write open parentheses, 11 Z plus three, close parentheses multiplied by open parentheses. 11 Z plus three, close parentheses as open parentheses, 11 Z plus three, close parentheses squared. And that's as factored as that one gets. We look at our answer choices and this matches with answer choice D while 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. 64y² − 16y + 1

Key Concepts

Perfect Square Trinomials

Factoring Polynomials

Prime Polynomials

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