1find all rational zeros of the polynomial and then find the...

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the Quadratic Formula, or other factoring techniques. (Enter your answers as comma-separated lists. Enter all answers including repetitions. If an answer does not exist, enter DNE.) P(x) = 12x^4 - 11x^3 - 18x^2 + 5x rational zeros x =

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00:01 For this question, we are going to find all the rational and irrational zeros of the given polynomial function using all of the tools that we know.

00:10 Let's start with the rational zeros theorem.

00:13 Our possible solutions are going to be positive or negative factors of six over factors of two.

00:32 So our possible things we're going to try are positive or negative 1 over 1, 2 over 1, 3 over 1, 6 over 1, 1 over 2, 2 over 2, which is 1, 3 over 2, and 6 over 2, which is 3.

00:55 Because all of the signs in our polynomial function are positive, one of our zeros must be negative.

01:03 So i'm going to start with one of my negatives.

01:07 And i've done some preliminary work ahead of time and know that negative 3 is one of the zeros that will work.

01:23 Bring down the 2.

01:25 Multiply.

01:27 Combine.

01:28 Multiply, combine, multiply, combine, multiply, combine, multiply, and we have a zero.

01:40 All right.

01:41 Now we have a cubic function that we need to try and find another value for.

01:57 But notice the ending number, the constant term, is a two.

02:04 That means we will no longer have to try positive or negative 3, positive or negative 6, positive or negative 3 halves.

02:13 We will try negative 1 1⁄2...

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