Ational Root Theorem Use the following function to answer...

ational Root Theorem Use the following function to answer parts a through c. f(x) = 2x^3+6x^2+5x+2 a. List all possible rational zeros. \pm1, \pm\frac{1}{2}, \pm2 (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each an b. Use synthetic division to test the possible rational zeros and find an actual zero. One rational zero of the given function is (Simplify your answer.)

Transcript

00:02 Okay, so we're given x cubed plus 5x squared minus 22x, minus 26.

00:11 And then the first part says given the rational zero theorem, okay, which basically says if i want to make a list of all the possible rational roots, i'm going to take all my factors of 26 and i'm going to divide them by all my factors of one.

00:40 So everything that divides into 26 is plus and minus 1, plus and minus 2, plus and minus 13, plus and minus 26.

00:50 And the only thing that goes into 1 is plus and minus 1.

00:54 So the only thing that i'm going to try in synthetic division, i'm going to do all my combinations.

00:59 I'm going to do 1.

01:01 I'm going to do plus and minus 2.

01:03 I'm going to do plus and minus 13, and i'm going to do plus and minus 26.

01:09 So this is going to give me all my rational zeros.

01:13 This does not include imaginary zeros.

01:17 This does not include radicals.

01:19 All it is is going to be my rational.

01:21 So for example, i would not try the number three.

01:24 I would not try the number five.

01:25 I wouldn't try one half.

01:27 I'm only going to try these numbers.

01:30 So then i'm going to take my equation, again, x cubed plus 5x squared minus 22x .x.

01:40 And i'm going to try some of these numbers.

01:42 So let's try one first.

01:44 And if it is a zero, then i should get zero as a remainder.

01:49 So i'm going to take all my coefficients, excuse me, 1, 5, negative 22, negative 26.

01:58 Let's try the number one first.

02:00 I'm going to bring down the 1.

02:01 1 times 1 is 1.

02:03 I'm going to put the 1 here.

02:04 I'm going to add 1 times 6 is 6.

02:08 Negative 22 plus 6 is negative 16 and you can kind of see from here obviously this is not going to be zero so one is not going to be one of my zeros so i'm going to try negative one next my goal is to get this down to a zero right here if i can so let's try negative one so bring down my one negative 1 this one's going to work i'm going to get zero so this is my remainder this is what i want so that means negative one that's good that gives me a zero um this is my remainder so when you get a zero remainder that means that is a root this is your constant term this is the coefficient for x this is the coefficient for x squared so you're left with x squared plus 4x minus 26 so your goal is to get it down from a cubic down to a quadratic that you can factor or do quadratic formula.

03:15 Since this is not factorable, i'm going to do quadratic formula.

03:19 Remember, quadratic formula is negative b plus or minus a square root of b squared minus 4a c over 2a...

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