The function below has at least one rational zero use this...

The function below has at least one rational zero. Use this fact to find all zeros of the function. h(x) = 5x^4 - 3x^3 - 13x^2 - 3x + 2 If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

Transcript

00:01 Alright, this function is 5x to the 4th minus 3x cubed minus 13x squared minus 3x plus 2.

00:12 And they're telling us that we know that function has at least one rational zero.

00:18 So i did the rational zeroes theorem where we had to take the factors of the constant and divide that by the factors of the leading coefficient.

00:42 So the factors of the constant, well constant's 2, so you have plus and minus 1 and plus and minus 2.

00:49 And the leading coefficient's 5, so that's plus and minus 1 and plus and minus 5.

00:53 So that means that the possible rational zeroes are 1 over 1, 1 over 5, 2 over 1, and 2 over 5.

01:10 So then i just kind of started.

01:17 I did synthetic division.

01:19 I put 1 in the box, pulled out my leading coefficients, making sure that i am in descending order by degree, bring down the 5, re -multiply, do what it says, re -multiply, do what it says, re -multiply, do what it says, re -multiply, and do what it says.

01:43 And i'm looking for a zero there.

01:46 So 1 is not one of the possible rational zeroes.

01:51 Let's try negative 1.

01:53 Leading coefficients, be very careful to make sure that you get all the signs right.

02:02 Bring down the 5, re -multiply, do what it says, re -multiply, do what it says, re -multiply, do what it says, re -multiply, do what it says.

02:13 Woo -hoo! i got a rational zero.

02:16 So i know one of these is x equals 1.

02:19 That's one rational zero.

02:22 I didn't try the fraction.

02:25 I just went to 2 just because it was a much nicer number.

02:30 And you can use the depressed polynomials equation if you would like.

02:35 So i did 5, negative 8, negative 5, and 2.