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 provides a list of possible rational zeros for a polynomial with integer coefficients by stating that any rational zero, expressed in lowest terms as p/q, must have a numerator p that is a factor of the constant term and a denominator q that is a factor of the leading coefficient. This theorem reduces the number of potential candidates that need to be tested when finding the zeros of the polynomial.

Factor Theorem

The Factor Theorem is a specific application of the Idea that a polynomial has a factor (x - c) if and only if the polynomial evaluates to zero at x = c. This fundamental theorem forms the basis for linking the zeros of a polynomial with its factors and is essential in the process of polynomial factorization.

Synthetic Division

Synthetic Division is an efficient algorithm for dividing a polynomial by a binomial of the form x - c. This method is particularly useful for testing potential rational zeros identified via the Rational Root Theorem and for simplifying the polynomial once a zero is confirmed, thereby reducing its degree.

Polynomial Factorization

Polynomial Factorization involves expressing a polynomial as a product of its linear factors and possible irreducible polynomial factors. Factoring completely often requires identifying the zeros, including rational zeros, and systematically breaking down the polynomial using division techniques and the Factor Theorem.

Transcript

00:01 Okay, for this problem, we are looking at the polynomial b of x is equal to 2x to the 6, minus 3x to the 5th, minus 13x to the 4th, plus 29x cube, minus 27x squared, plus 32x ,000, minus 12.

00:31 Now for this one, we are going to want to look according to the rational values now.

00:38 We look at all the potential divisors plus or minus of negative 12, which would be 1, 2, 3, 4, and 12, divided by all potential divisors of the leading coefficient, which should be plus and minus 1 or 2.

00:57 With all fractions, these numerators, these are nominators, and we can start testing it out.

01:03 And let's actually use the synthetic division technique that we ended in a section a bit early, but this problem is quite heavy.

01:10 So take an extraordinary amount of time guessing to the solutions.

01:15 So let's start using our technique.

01:17 Let's guess two.

01:19 So we'll write our two right here.

01:21 And then we'll use our synthetic division technique.

01:24 This entirely corresponds to long division of polynomials.

01:27 You can look at the section if you forgot and see how these operations are exactly the same as the operations there.

01:35 Maybe just written a little differently okay so we carry the two down here we multiply two times two we get four we add we get one now we multiply these two we get two we add we get negative 11 we multiply these two we get negative 22 we add downward we get seven we multiply this and that we get 14 now we add going down as usual, we get negative 13.

02:13 We multiply again, negative 26, we add down, we get six, then we multiply again, we get 12, and they cancel that and get 0.

02:24 What does that mean? well, i said this correspond exactly to long division.

02:28 So if we get to the very last term and we get a zero, it means the remainder was 0, and that means this polynomial has a term like this, x minus 2, what we guessed was a real 0 of 2.

02:46 And this corresponds to having a real 0 of 2.

02:49 And since it corresponds to long division, can you guess what all these are? well, they would describe a quotient polynomial.

02:56 Quotient meaning you multiply this by that, you add some remainder, and you will get your original polynomial.

03:04 2x to the 6th.

03:07 We're using these numbers.

03:09 Okay, i just started to complete the original polynomial.

03:13 I'm going to do that.

03:14 This tells you new information.

03:17 If you divide by this, you get 2x to the fifth and then the leading coefficient for the next term is just 1 plus x to the 4th.

03:25 Leading coefficient to the next term is negative 11.

03:28 To the next term, it's 7, x squared, to the next term it's negative 13.

03:34 X.