In Exercises 1–22, factor each difference of two squares. Assume ...

Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers.

a² - (b - 2)²

Accepted Answer

Hello, everyone. The following expression is a special product called the difference of two squares. We're gonna factorize it. The expression is P squared minus the parentheses V minus nine squared. Our answer choices are a parentheses, P minus V plus nine multiplied by the parentheses. P plus V minus nine B, the parentheses, P minus V plus nine multiplied by the parentheses. P minus V minus nine C, the parentheses, P minus V plus three multiplied by the parentheses. P plus V minus three D, the parentheses P minus V plus three multiplied by the parentheses. P minus V minus three. Going back to the expression we're given P squared minus parentheses, V minus nine, close parentheses squared here to have the difference of two squares. We're gonna treat the parentheses V -9 as if it's one single term. I'm not going to remove those until a future step. Recall that when we have the difference of two squares that we're gonna factor our original binomial into two other binomials. So I opened two sets of parentheses. They are currently empty as we wait to factor this, the first term in each set of parentheses will come from the square root of the first term in the difference of two squares. So the square root of P squared is P and that goes in the first spot in both sets of parentheses. Next, I want to find the last term in both new binomials. And that's found by the square root of the second term in the difference of two squares. In this case, they'll be the square root of the parentheses V - squared. This will just leave us with V -9. And I'm gonna put that in, in parentheses in both of my binomials. Next, I recall that I need to have a plus sign and a minus sign one in each set of my parentheses. So in my first set of parentheses, I will have P minus the parentheses, V minus nine, closed parentheses, closed parentheses, multiplied by open parentheses. P plus the parentheses, V minus nine, clothing, both sets of parentheses. One more thing I can do here is I want to distribute the sign in front of the parentheses into the parentheses V -9. So in our first binomial, I'm gonna reopen my parentheses. I have P and then I have a minus sign. So I'm gonna distribute that into the parentheses as a negative one. So negative one times V gives me negative V negative, one times negative nine, gives me plus nine closed parentheses. Opening my next set of parentheses P and then I have a plus sign And since that's a plus sign, it will not change the values within the V -9 parentheses. So I can drop those parentheses and just rewrite it as V -9 and close parentheses. So as a whole, I have open parentheses. P minus V plus nine, closed parentheses multiplied by open parentheses. P plus V minus nine, closed parentheses. I have nothing else here. I can factor. So this is my solution and it's answer choice. A, have a nice day.

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. a² - (b - 2)²

Key Concepts

Difference of Squares

Factoring

Variable Exponents

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