In Exercises 39–44, factor by introducing an appropriate substitu...

Question

In Exercises 39–44, factor by introducing an appropriate substitution.(x + 1)² + 8(x + 1) + 7 (Let u = x+1.)

Accepted Answer

Hello, everyone. We're gonna perform factorization using substitution. The expression we're given is open parentheses. X plus two, closed parentheses squared minus three, multiplied by open parentheses. X plus two closed parentheses minus 28. When we do our substitution, we are going to let U equal X plus two. Our answer choices are a open parentheses. X -5 closed parentheses multiplied by open parentheses. X -6, closed parentheses. B open parentheses, X plus five closed parentheses multiplied by open parentheses. X plus six, closed parentheses. C open parentheses, X plus five closed parentheses, multiplied by open parentheses. X minus six closed parentheses. D open parentheses, X minus five, closed parentheses multiplied by open parentheses. X plus six, closed parentheses. I'm going to rewrite the expression as it is given to us originally. So open parentheses, X plus two, close parentheses squared minus three, multiplied by open parentheses, X plus two, closed parentheses minus 28. We were told to let U equal X plus two. So I'm gonna put you into this trinomial where the parentheses X-plus 2 are now. So this will give me U squared minus three, U minus 28. This looks a lot more like the trinomial that we're used to factoring. So now we're going to use that same format that we have used all along. I recall that when I factor a trinomial, it's going to result in two binomials. So I'm gonna open two sets of parentheses next to each other. The first term in, in my parentheses needs to multiply to my first term in my trinomial. So to get U squared, I'm gonna have U times U so U is the first term in each set of parentheses. Next I need to find two numbers that not only multiply to negative 28 but add to a negative three knowing that it multiplies to a negative number. I know they need to have opposite signs. I'm gonna list all my combinations that multiply to negative 28 off to the side. So negative one and 28 1, negative 28 -2 and 14, two and negative 14 negative four, positive seven, positive four, negative seven. And I believe that's all the combinations I'm gonna see if any of those or which one adds up to a negative three, it will not be negative one and 28 that does not work, it will not be one and negative 28. When I do negative two plus 14, that is 12, that is not negative three, same with two and negative 14, negative four and positive seven gets me to a positive three. But I want a negative three. So 4 and - are the numbers I want to work with. So I'm gonna put those in my parentheses as plus four in the first binomial -7 in my second binomial. So right now I have open parentheses, U plus four, closed parentheses, multiplied by open parentheses. U minus seven, closed parentheses. Typically this is where we would stop our factorization, but now that we have this factored, we're gonna substitute back our original X-plus 2. So rewriting this, I have open parentheses and instead of you, I'm now putting X plus two plus four, closed parentheses, Open Parentheses, X-plus -7, closed parentheses. Now within my parentheses, I have like terms, I'm gonna combine my like terms and then we'll see if there's any other factorization that can be done. So, open parentheses in the first set of parentheses, the X has nothing to be changed or combined with. So that stays X two plus four is six. So our first set of parentheses is X plus six. Opening the next set of parentheses. Again, the X doesn't change. I have two plus negative seven, which is negative five. So my second parentheses is X minus five. Looking at this, I do not see anything else that can be factored or combined. So this is my solution and I have open parentheses, X plus six, closed parentheses multiplied by open parentheses, X minus five, closed parentheses. Recall that with multiplication, our factors don't have to be listed in the exact order to be equivalent. What I mean is this is answer choice D because our terms match, we have X plus six multiplied by X minus five. But in answer choice, D they write it as open parentheses, X minus five, closed parentheses multiplied by open parentheses X plus six. But they do mean the same. Have a nice day.

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In Exercises 39–44, factor by introducing an appropriate substitution.(x + 1)² + 8(x + 1) + 7 (Let u = x+1.)

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