Video: Using Synthetic Division to Find Zeros of Polynomials

One of the zeros of the function 𝑓(𝑥) = 𝑥³ − 4𝑥² − 17𝑥 + 60 belongs to the set {2, 3, 4}. Using synthetic division, find all zeros of 𝑓.


Video Transcript

One of the zeros of the function 𝑓 of 𝑥 equals 𝑥 cubed minus four 𝑥 squared minus 17𝑥 plus 60 belongs to the set two, three, four. Using synthetic division, find all zeros of 𝑓.

So here, we have our function in the possibility of the zeros: two, three, or four. And we know that only one of these is one of the zeros of the function. A zero of a function will divide evenly into the function. So we need to take these numbers and use synthetic division to decide if it’s a zero of the function or not.

Let’s begin with two. So we write it outside like this. And then, we use all of the coefficients and the constants and we make sure that’s in decreasing order. So we have one, negative four, negative 17, and then 60. So for synthetic division, we bring down the first number which is a one. And now, we take one times two. And we’ll write it here. And one times two is two. And now, we add the columns together. So negative four plus two is negative two.

And now, we’ll repeat the process. Negative two times two is negative four. And negative 17 plus negative four is negative 21. And then, negative 21 times two will be negative 42 and 60 plus negative 42 would be 18. So the way to read these numbers is all the way to the right, which is the 18 is our remainder. That’s what’s left over. And then, negative 21 would be the constant number, negative two would be the coefficient with 𝑥, and one will be the coefficient with 𝑥 squared. And then we’d keep going if there would have been more numbers.

So since there is a remainder, that means that two did not divide evenly into our function. So two is not one of our zeros. So now, let’s try three. So using the same process of synthetic division, we bring down our one and one times three is three. And negative four plus three is negative one and negative one times three is negative three. And negative 17 plus negative three is negative 20. And now, negative 20 times three is negative 60 and 60 plus negative 60 is zero. This means we have a remainder of zero. That’s great. That means three went into our function evenly. So three worked.

Now, let’s try four just to be safe, even though one of them should have only worked. So we bring down our one. And one times four is four and negative four plus four is zero. And then, zero times four is zero. And negative 17 plus zero is negative 17. And negative 17 times four is negative 68. And 60 plus negative 68 is negative eight. So there would be a remainder of negative eight. So it would not divide evenly. Therefore, three must be our zero.

Next, it says find all zeros of 𝑓. So, if we know that three is one of them, we can use it to help find the other ones. So we know that zero was our remainder, negative 20 was a constant, negative one was a coefficient with 𝑥, and one was the coefficient with 𝑥 squared. So we have the function 𝑥 squared minus 𝑥 minus 20. And we can set it to zero and factor to find the other zeros.

So what are two numbers that multiplied to be negative 20 and add to be negative one? That would be negative five and positive four. So to find the answer though, the zeros of the function, we need to set each factor equal to zero. So to begin with 𝑥 minus five, to solve for 𝑥, we need to add five to both sides of the equation. And we have that 𝑥 is equal to five. For the next one, we need to subtract four from both sides of the equation and we find that 𝑥 is equal to negative four.

So our zeros for this function would be negative four, three, and five.

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