In Exercises 69–82, factor completely.10y⁵ – 17y⁴ + 3y³

Question

Accepted Answer

Hello, today we're gonna be factoring the given trinomial. So what we are given is 12 Y to the fifth power minus 23 Y to the fourth plus 10 Y Q. Now, before we factor this using the trial and error method, let's go ahead and check to see if we could factor out a greatest common factor. So if we take a look at the coefficients, 1223 and 10 do not have any common multiples with each other. But in each term, notice how there is at least three instances of Y in each term. That means we can go ahead and factor out a y cubed from the given trinomial. So if we factor out A Y cubed, what we end up with is Y cubed times the quantity 12 Y squared minus 23 Y plus 10. And now in order to factor this, we're gonna go ahead and use the trial and error method. So what we're going to do is list out the factors of the first term and the last term and the factors of the first term which I'm going to call the P factors are any possible factors made up of only positive integers. And for the last term, which I'm going to call the Q factors, these factors can be made up of any combination of positive or negative numbers. So what are going to be the possible factors for 12 Y squared? Well, we can go ahead and multiply Y with 12 Y, we can multiply four Y with three Y and we can also try six Y with two Y. And what about the factors for the last term? Well, the last term is positive, which means we can multiply two negative numbers together to get a positive number or we can multiply two positive numbers together. So the possible factors for 10 is going to be one in 10 or negative, one, negative 10, we can also try five and two or negative five and negative two. So now that we've listed out all the factors, what I'm going to do is create two pairs of parentheses and inside these pairs of parentheses, the first terms are going to be the P factors and the last terms are going to be the Q factors which can be positive or negative. So now what we wanna go ahead and do is pick any combination of P and Q factors to test. So let's go ahead and try four Y with three Y And let's go ahead and try positive five and positive two. So if we put this into our P Q form, what we end up with is four Y plus five times the quantity three y plus two. And in order to check to see if this is the correct factorization, we're going to multiply the very first with the very last term together. And we're gonna multiply the two middle term, two terms together and add up their products. So four Y times two is going to give us eight Y and five times three Y is going to give us positive 15 Y and so 15 Y plus eight Y is going to give us 23 Y. Now this does not match up with the original middle term which is going to be negative 23 Y. So this is not going to be the correct factorization but we got very close to negative 23 Y we just ended up getting a positive number. So let's try the same combination of factors but let's go ahead and switch the signs. So instead of using positive numbers, let's go ahead and try the negative numbers. So we're gonna try four Y minus five times. The quantity three Y minus two, four Y times negative two is going to give us negative eight Y and negative five times three Y is going to give us negative 15 Y a negative eight Y minus 15 Y is going to give us negative 23 Y which matches up with the middle term of the trinomial. So that means that four Y minus five times three Y minus two is going to be the factored form of the trinomial. But be careful because recall that we factored out A Y cubed in the beginning of the problem. So let's go ahead and combine all of our factors together that is going to give us Y cubed times the quantity four Y minus five times the quantity three Y minus two. And this is going to be the completely factored version of the given trinomial. And with that being said, the answer to this problem is going to be D oh I'm sorry, the answer to this problem is going to, yeah, it's going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 69–82, factor completely.10y⁵ – 17y⁴ + 3y³

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Polynomial Degree and Terms

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