Find all rational zeros of each function. ƒ(x)=8x^4-14x^3-29x^2-4...

Question

Find all rational zeros of each function. ƒ(x)=8x^4-14x^3-29x^2-4x+3

Accepted Answer

Hello, today we're going to be solving for the rational roots of the following polynomial function. So what we are given is F of X is equal to six X to the fourth minus X cubed minus 37 X squared plus 62 X plus 24. Now, the first thing we're going to need to do is find all the possible rational routes for the following polynomial function. And in order to do that, we can list out all the possible factors for the first term. And for the last term, the factors of the last term are going to be called the P factors and the factors for the first term are going to be called the Q factors. So what are the possible factors for 24? Well, 24 has quite a bit of possible factors, but the full list is going to be 123468, 12 and 24. And now we need to list out the factors for the Q factors. Well, the possible factors for six are going to be 123 and six. Now, what we're going to do is we're going to use the possible rational root theorem. And we're going to list all the factors as P over Q. Now that's going to give us this large list, which is going to be 123468, 12 and 24 all over the Q factors which are going to be 123 and six. Now, if we simplify this list, we're going to get plus or minus +11 half, one third, 1/6 +22 3rd, 3, 3/2, 4, 4/3, +68, 8/3, 12 and 24. So what we've just listed out are all the possible values for the rational roots of our polynomial function. And now we want to apply the rational root theorem. The rational root theorem states that if we take any possible rational route, plug it into our function and it give us the value of zero, then that is going to be a rational route for the polynomial function. Now, if you use a calculator and you plug in the values of negative 1/3, negative 2, 3/2 and four, these are all going to give the output of zero. And any other possible route plugged into the function is not going to give the value of zero. So that means that these four values here are going to be the possible routes for the polynomial function. 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 approach this problem. And I'll go ahead and see you while in the next video.

Find all rational zeros of each function. ƒ(x)=8x^4-14x^3-29x^2-4x+3

Key Concepts

Rational Root Theorem

Synthetic Division

Polynomial Functions