Factor each polynomial completely. See Example 6.6a² - 11a + 4

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to rewrite the following expression by factoring it completely. Our expression is 10 R squared minus 26 R plus 12. Our answer choices are answer choice A three multiplied by the quantity of five R minus two multiplied by the quantity of R minus three. Answer choice B two multiplied by the quantity of five R minus three multiplied by the quantity of R minus two. Answer choice C two multiplied by the quantity of three R minus five multiplied by the quantity of R minus two. And answer choice D three multiplied by the quantity of two R minus five multiplied by the quantity of R minus three. All right. So looking at our expression, the first thing we want to do is look at the three terms of our expression R squared, A negative 26 R and A positive 12. And see if they have any common factors that we can factor out in front. So let's look at the coefficients 1st 10, negative 26 and the constant term 12 do 10. We'll just use, we'll just look at positive 26 for now and 12 have any common factors. They do, they're all divisible by 2. 10 equals two, multiplied by 5, 26 equals two, multiplied by 13 and 12 equals two, multiplied by six. Now, between the five, the 13 and the six, those do not share any common factors. So all we can take out is that two from our numbers. How about our variables? Do all three terms share any variables. They do not, the 12 does not have an R attached to it. So we can't factor out any variables. So just the two, the two is multiplied by open parentheses. Now we are going to divide all three of our terms by two. And that will be what's inside of our parentheses. 10 R squared divided by two leaves us with five R squared, negative 26 R divided by two, leaves us with negative 13 R and a positive 12 divided by two leaves us with plus six. All right. So we have got this partially factored, but now we want to focus on what's inside of the parentheses and see if we can factor that any further and we can do that with the reverse of the foil process. We can bring down this two and then we have two open sets of parentheses that it's being multiplied by. We want to begin by asking ourselves what multiplied by what gives us five R squared. Those are going to be our firsts. And for five R squared. Well, one of them is going to be a five R and the other is going to be an R. Next. We want to ask ourselves what multiplied by what gives us a six positive six. Those are going to be our lasts. And we've got options for a positive six. For a positive six, we could have one multiplied by six. We could have negative one multiplied by negative six. We could have two multiplied by three. We could have negative two multiplied by negative three. We could also have three multiplied by two. We could have a negative three, multiplied by a negative two. And we could have a six multiplied by one or a negative six multiplied by a negative one order does matter for the lasts. So we've got a lot of options there. How do we narrow it down? Well, we need to be able to discern for our outsides when we multiply those and add those to our insides together. Those need to add up to a negative 13 R. Well, because they need to add to equal a negative 13 R that gives us a clue that our factors for six are going to be negative. So we can eliminate all of the positive multiplied by a positive and that just leaves four alternatives. Now we know that we're going to multiply it five by something and add it to something else and it needs to add up to a negative 13. Well, to get close to that, we could consider multiplying the five by two. So we can do some guess and check here, we've got one position where we would have a two being multiplied by the five. That's what the factors with negative three and negative two. So if we had five R minus three in one set of parentheses multiplied by the quantity of R minus two. Let's foil this out and see if it works. So first five R multiplied by R five R squared, we know that one's good for our outsides five R multiplied by negative two. That's negative 10 R, our insides negative three multiplied by RS negative three R and our lasts negative three, multiplied by negative two is a positive six. Now, this one looks like it's going to work out. Sometimes you guess correctly, sometimes you have an educated guess to get it correct. And sometimes you're just plugging in all of the different solutions to see what you get. So if we simplify this by combining like terms in the middle, negative 10 R minus three, R is the negative 13 R that we were looking for. And our first and our last we know that those work. So this is our factored solution. We have the two multiplied by the quantity of five R minus three multiplied by the quantity of R minus two. Can't forget that two hanging out in front. So we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.6a² - 11a + 4

Key Concepts

Factoring Polynomials

Quadratic Polynomials

Zero Product Property

Watch next