Rational Zeros Find all rational zeros of the polynomial, and write the polynomial in factored f

Rational Zeros Find all rational zeros of the polynomial,...

Key Concepts

Rational Root Theorem

The Rational Root Theorem is used to identify a list of possible rational candidates for the zeros of a polynomial with integer coefficients. It asserts that any rational zero is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This concept is fundamental in narrowing down the potential rational zeros of the polynomial.

Synthetic Division or Polynomial Division

Once potential zeros are proposed using the Rational Root Theorem, synthetic division or polynomial division is applied to test whether a candidate is indeed a zero. These division methods help factor the polynomial by reducing its degree, thereby simplifying the task of finding the remaining zeros.

Factoring Polynomials and the Zero Product Property

After confirming a zero, the polynomial can be factored into linear or irreducible components. The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero, which provides the basis for solving the polynomial equation once it is expressed in factored form.

Transcript

00:01 Okay, in this video we are going to factorize this polynomial, and we are going to find its rational zeros.

00:08 So first notice the degree.

00:10 This may be a degree five polynomial, which could be a little intimidating, but we're going to do the exact same procedure that we've been doing on the degree four polynomials and degree three polynomials.

00:22 So first thing we're going to do is look at this constant term, which is 36, find its factors, and look at the factors of the leading co -conumials.

00:31 Which is just one, which means all of our possible zeros are going to be integers.

00:38 So the factors of 36 are plus or minus 1, 2, 3, 4, 6, 9, 12, 18, and 36.

00:47 So first thing we're going to do is our division 1x to the 5th plus 3x to the 4th minus 9x cubed minus 31 x squared plus 0 x, don't forget the 0, plus 36.

01:10 Okay? now, of course, we got to drag the 1 down.

01:15 So our first value that we're going to try out, let's try out a positive 1.

01:21 Okay? 1 times 1 is 1 .3 plus 1 is 4.

01:25 1 times 4 is 4 minus 9 is negative 5.

01:31 1 times negative 5 is negative 5 minus 31 will give you negative 36 times 1 is still negative 36 0 minus 36 is negative 36 and multiply that again by 1 still negative 36 nothing's changed okay now 36 minus 36 we'll give you 0 which means 1 is a 0 of the polynomial so we have right here, x minus 1 times x to the fourth plus 4x cubed minus 5x squared minus 36x minus 36.

02:35 Okay.

02:37 So we're going to do our division again with that polynomial.

02:44 So 1, 4, negative 5.

02:50 Negative 36 and negative 36.

03:01 Now we've got to try out another value.

03:03 Let's do three.

03:07 So you're going to drag the one down.

03:11 Three times one is three.

03:16 Four plus three is seven.

03:22 Three times seven is 21...