Rational Zero Theorem. Solving Polynomial Equations So far, we have learned how to solve polynomial equations by graphing and by factoring. Graphing is.

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Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

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Presentation transcript:

Rational Zero Theorem

Solving Polynomial Equations So far, we have learned how to solve polynomial equations by graphing and by factoring. Graphing is nice because you can see the zeros of the function, but if your function has imaginary zeros, you can’t see those on the graph. Factoring is nice because you can find both real and imaginary zeros…but factoring can sometimes be difficult based on the polynomial you have.

In this lesson, We will work to make factoring these difficult polynomials a little simpler. We will use the coefficients of the terms of our polynomial equation to help us identify possible solutions.

Consider x 3 – 7x + 6 = 0 Using what we have learned so far about factoring, we can’t determine the factors. I will tell you that in factored form, the equation would be (x + 3)(x – 2)(x – 1) = 0. The solutions of this equation are x = -3, 2, 1. Notice that if you multiply the solutions you get -6. The solutions to a polynomial equation are related to the quotient of the constant (in our case 6) and the leading coefficient (in our case 1).

The Rational Zero Theorem Suppose we have the polynomial equation a n x n + a n-1 x n-1 + … + a 1 x + a 0 = 0 We can generate a list of all possible rational roots (or solutions) of our polynomial equation by computing, where p must be a factor of a 0 and q must be a factor of a n.

For example, Our polynomial equation earlier was x 3 – 7x + 6 = 0 To compute all possible rational zeros, we can do, where p must be a factor of 6 (our constant) and q must be a factor of 1 (our leading coefficient). In this case,. Divide the numerators and denominator to get all possible integer solutions:

If you’ll remember, We said the solutions were -3, 2, and 1. Notice that all 3 of these are in our list of possible solutions! Keep in mind that imaginary solutions and irrational solutions will not show up in our list. Because of this, we will usually use the Rational Zero Theorem in conjunction with another tool to locate ALL of our zeros.

Let’s try another example. List all possible rational zeros of 3x 3 – x 2 – x + 4 = 0. First, set up your ratio: Next, divide to get your list of possible rational zeros. It is okay to get fractions, since they are rational numbers!

You try this one. Click on the next slide to check your answers. List all possible rational zeros of 2x 3 – 2x 2 – 5x + 10 = 0.

How well did you do? Check your work! Ratio: List of possible rational answers:

Now let’s talk about how to use the Rational Zero Theorem to help us solve a polynomial equation!

Let’s solve x 3 - 2x 2 + 5x - 10 = 0. First, use the Rational Zero Theorem to identify possible rational solutions: The list would be: Begin testing possible zeros. When substituted into your equation, a zero will give you zero: ▫Test x = 1: (1) 3 – 2(1) 2 + 5(1) – 10 = -6…so not a zero ▫Test x = -1: (-1) 3 – 2(-1) 2 + 5(-1) – 10 = -18…so not a zero ▫Test x = 2: (2) 3 – 2(2) 2 + 5(2) – 10 = 0…so x = 2 is a solution!!!

x 3 - 2x 2 + 5x - 10 = 0, continued Once we find one solution, we can use synthetic division to help us find the rest: Let’s solve the resulting polynomial equation: x = 0 x 2 = -5

x 3 - 2x 2 + 5x - 10 = 0, continued Our equation has three solutions:  x = 2  x =  x = - Notice that the graph of our equation confirms our results— only one real solution is shown Having the rational zero theorem helps us solve polynomial equations that we wouldn’t be able to solve if we only depended on graphs!

Let’s solve 2x 3 – x 2 + 2x – 1 = 0. First, use the Rational Zero Theorem to identify possible rational solutions: The list would be: Begin testing possible zeros. When substituted into your equation, a zero will give you zero: ▫Test x = 1: 2(1) 3 – (1) 2 + 2(1) – 1 = 2 ▫Test x = -1: 2(-1) 3 – (-1) 2 + 2(-1) – 1 = -6 ▫Test x = 0.5: 2(0.5) 3 – (0.5) 2 + 2(0.5) – 1 = 0…so x = 0.5 is a solution of our equation!

2x 3 – x 2 + 2x – 1 = 0, continued Once we find one solution, we can use synthetic division to help us find the rest: Let’s solve the result. Notice we get imaginary answers:  2x = 0  2x 2 = -2  x 2 = -1  x = i, -i

2x 3 – x 2 + 2x – 1 = 0, continued Our equation has three solutions:  x = 1/2  x = i  x = -i Notice that the graph of our equation confirms our rational result This example again shows why we can’t just rely on a graph for our solutions…we wouldn’t be able to find imaginary solutions!