Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. x^3-7 x-6

Name: Problem 66, Chapter 0: Calculus an Applied Approach
Uploaded: 2021-06-18T21:55:27-08:00
Duration: 4 min 39 s
Channel: Christian Otero
Description: Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. x^3-7 x-6

Use the Rational Zero Theorem as an aid in finding all real...

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of lower-degree polynomials. This concept is important for finding all zeros of the polynomial, since once the polynomial is factored, the zeros can be found by setting each factor equal to zero. Factorization, coupled with the Rational Zero Theorem and synthetic division, provides a systematic approach to solving polynomial equations.

Synthetic Division

Synthetic division is an efficient method for dividing polynomials by linear factors of the form (x - c) and is especially useful when testing potential roots identified by the Rational Zero Theorem. It streamlines the process of determining whether a candidate is indeed a zero of the polynomial and assists in reducing the polynomial's degree for further analysis.

Polynomial Zeros

A polynomial zero (or root) is a solution to the equation formed when the polynomial is set equal to zero. Finding the zeros of a polynomial is a central task in algebra because the zeros indicate where the polynomial intersects the x-axis, and they help in understanding the behavior and factorization of the polynomial.

Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial with integer coefficients has any rational zeros, then each zero can be expressed in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem helps narrow the list of possible rational candidates that can be tested as actual roots of the polynomial.

Transcript

00:01 So in this video, we want to look at the polynomial x cubed minus 7x minus 6 and use the rational root theorem or the rational zero theorem to help find all the real zeros of the polynomial.

00:14 So what the theorem tells us, so this theorem tells us that if we consider p as the factors of our constants and q as the factors of the leading term, we know all the factors.

00:39 Rational zeros are of the form p divided by q so we just want to find these p's in these cues and we can go ahead and look at our two terms so we have p is the factors of the constant which in this case is negative six and so we know p can be plus or minus one plus or minus two plus or minus three and plus or minus six and so q is just the factors of our leading term which in this case is one so we know q can only be plus or minus 1.

01:14 And all of the forms of our values are p divided by q.

01:20 And since q is only plus minus 1, we know all the possible zeros are the box circled and green.

01:26 So these are all p over qs since q is just plus or minus 1.

01:36 And so we can go ahead and start going down a list that's starting from positive 1 to negative 1 to positive 2 to negative 2.

01:43 Until we go ahead and find a form of the equation that's easy enough to factor without using these synthetic division methods.

01:52 So by using synthetic division, we can, one, evaluate the function, and two, also factor it if it is a zero.

02:03 So if we go ahead and do x is equal to one first, so let's do this in black actually, x equals one.

02:11 We know if we go ahead and rewrite our polynomial, we have x cubed minus 7x minus 6.

02:18 So we, and it's equal as zero, can't forget that part.

02:23 So we want to write our coefficients, we're just 1, 0, negative 7, negative 6...

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