Factor each polynomial. See Example 7. 6(4z-3)^2+7(4z-3)-3

Question

Accepted Answer

Hello, today we're going to be factoring the given polynomial. So what we are given is six times the quantity two X plus three squared minus five times the quantity two X plus three minus six. Now notice that two X plus three pops up more than once in this given polynomial to make factory in this polynomial easier. We can go ahead and make a substitution for two X plus three. So if we let a variable U equal to two X plus three, once we make the substitution, the polynomial is gonna change to six U squared minus five U minus six. And this is going to be of the form of a fact, Herbal Trin Amiel in order to factor this binomial, we're going to use the trial and error method. So we're going to need the factors of the first term and the last term, the factors of the first term which is six is going to be one in six or two and three. And the factors of the last term which is negative six is going to be one in negative six, negative one in 62 and negative three or negative two. And three. And so when we choose a combination of these factors, factoring this try no meal is going to give us an answer of the form A U plus B times A U plus B. So A is going to be the factors for the first term and B is going to be the factors for the last term. So what combination of factors should we choose and test to see whether this is going to be fact herbal or not? Well, let's go ahead and choose two and 3 to be the factors for a and let's go ahead and choose to a negative three to be the factors for B. So if we plug this into our trial and error, what we get is three U plus two times two, U -3. Now we need to check to make sure that this is going to factor or um foil into our original try no meal. If we foil three, U plus two and two, U plus three, we're going to multiply three you into to you and negative three and we're going to multiply two into two, um negative three. This is going to give us three U times to you which is six U squared plus three U times negative three, which is going to give us negative nine you two times to you, which is going to give us for you and two times negative three, which is going to give us negative six once we combine like terms negative nine U plus four U is going to give us negative five you, which gives us six U squared minus five U minus six. So this confirms that three U plus two times two U minus three is going to be the factored form of the original Tryin Amiel. Now keep in mind we made a substitution for you earlier in the problem, the substitution for you is going to be two X plus three. So we're going to need to re substitute the value of you back into these factors and replacing you with two X plus three is going to give us three times the quantity two X plus three plus two times two times the quantity two X plus three minus three. Now, in order to simplify this, we're going to distribute three into two X plus three and we're going to distribute two into two X plus three. This is going to give us six X plus nine plus two times four, X plus six minus three. We're gonna go ahead and combine like terms by 1892 together and adding six and minus three together. And this is going to give us six X plus Times four X -3. And this is going to be the completely factored form of the original expression. And with that being said, the answer to this problem is going to be C so I hope this video helps you in understanding how to factor the original expression. And I'll go ahead and see you all in the next video.

Factor each polynomial. See Example 7. 6(4z-3)^2+7(4z-3)-3

Key Concepts

Factoring Polynomials

Substitution Method

Quadratic Form

Watch next