In Exercises 83–92, factor by introducing an appropriate substitu...

Question

In Exercises 83–92, factor by introducing an appropriate substitution.

3(x+1)² − 5(x+1) + 2 (Let u = x+1)

Accepted Answer

Hello, everyone. We are gonna use the appropriate substitution to factor the given trinomial. The trinomial we are given is six, multiplied by open parentheses. X -4, closed parentheses squared -77, multiplied by open parentheses, X -4, closed parentheses plus 121. Our answer choices are a open parentheses. 6 X -35 closed parentheses multiplied by open parentheses. X -15 closed parentheses. B open parentheses. Six X plus 35 closed parentheses multiplied by open parentheses. X plus 15, closed parentheses. C open parentheses. 6 X -11, closed parentheses, multiplied by open parentheses. X -11 closed parentheses. D open parentheses, X minus 11 closed parentheses multiplied by open parentheses. Six X plus 11, closed parentheses. Going back to our original trinomial as the directions tell us they want us to do substitution. What do they want us to substitute? Well, they want us to put one variable in for the binomial X -4. So I'm gonna say that X -4 is gonna be replaced by the letter U. So now I'm going to rewrite my trinomial and instead of the parentheses, X -4, I'm gonna substitute in U. So I have six U squared -77 U Plus 121. Now, I want to factor this trinomial. I recall that when I factor a trinomial, I'm gonna ultimately try to get it to be two binomials looking quickly at this to see if there is a greatest common factor I can take out of each term there does not appear to be. So I'm gonna open two sets of parentheses, they are empty and next to each other. And I'm gonna start breaking this down. It might be a little bit of guess and check, but we'll get there. The first spot in both binomials needs to multiply to the first term in our trinomial. So I need two values that will multiply to six us squared. I'm gonna go with six U and you because I know that will multiply back to six U squared. Next, looking at my signs, I know my two values that I multiply to get have to both be negative because the middle term is negative. So it has to multiply to a positive add to a negative, which means both parentheses have a minus sign. Now, I'm thinking of all of my options that are gonna going to multiply to 121 and I'm gonna test them. So I first one that comes to my mind is 11 and 11. Um So I'm gonna try that, I'll put them in both of my parentheses and then I'm gonna check it. So if the factorization is the parentheses, six X minus 11, multiplied by the parentheses, U minus 11, our middle term will tell us whether or not we have this correct. So inner would give us -11 U. Outer would give us -66 U. And when we add those together, that'd be negative 77 U, that's what we want our middle term to be. So this actually does check, I honestly expected to have to check one more set. But right now our factorization is open parentheses. 6 U -11, closed parentheses multiplied by open parentheses. U -11 closed parentheses. Now, we can't say we're done because we are not, we need to substitute back the X -4 for you. So in my first set of parentheses, I have opened parentheses six and I'm gonna multiply that by the parentheses, X -4, then closed parentheses -11, closed parentheses. And the second binomial, I opened my parentheses. I'm gonna put in X -4 and then -11, close parentheses. Now that I've removed the U from our binomial, I'm going to distribute the six into the X -4 in my first binomial. So open parentheses, six multiplied by X is six, X six, multiplied by negative four is negative 24. Still minus 11 closed parentheses, open parentheses, X -4 -11 closed parentheses. In the next step, I'm going to combine all of my like terms that are inside the parentheses. So in the first set of parentheses, the six X does not combine with anything but I have negative 24 negative 11 which should be negative 35 closed parentheses multiplied by next set of parentheses. The X has nothing to be combined with 4 -11. I'm sorry, negative 4 -11 is negative 15 closed parentheses. So our factorization here is open parentheses. Six X minus 35 closed parentheses multiplied by open parentheses, X minus 15 closed parentheses. And that's answer choice. A have a nice day.

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In Exercises 83–92, factor by introducing an appropriate substitution. 3(x+1)² − 5(x+1) + 2 (Let u = x+1)