Rational Root Theorem

1. Rational Root Theorem Possible Rational Roots What are the possible rational roots of Factors of the constant term, 6, are Factors of the lead coefficient, are Possible rational roots are

2. Possible Rational Roots • What are the possible rational roots of • Factors of the constant term, 8, are • Factors of the lead coefficient, 2, are • Possible rational roots are

3. Always keep in mind the relationship among zeros, roots, and x-intercepts. The zeros of a function are the roots, or solutions of the equation . The real zeros, or real roots, are the x-intercepts of the graph of .

4. Finding Zeros of a Polynomial Function • Use the Rational Zero Theorem to find all possible rational zeros. • Use Synthetic Division to try to find one rational zero (the remainder will be zero). • If “n” is a rational zero, factor the original polynomial as (x – n)q(x). • Test remaining possible rational zeros in q(x). If one is found, then factor again as in the previous step. • Continue in this way until all rational zeros have been found. • See if additional irrational or non-real complex zeros can be found by solving a quadratic equation.

5. Finding Rational Zeros Find the rational zeros for Find Possible zeros are+ 1, + 2, +4, +8 So which one do you pick? Pick any. Find one that is a zero using synthetic division...

6. Let’s try1. Use synthetic division 1 1 1 –10 8 1 2 –8 1 2 –8 0 1is a zero of the function The depressed polynomial is x2 + 2x – 8 Find the zeros ofx2 + 2x – 8by factoring or (by using the quadratic formula)… (x + 4)(x – 2) = 0 x = –4, x = 2 The zeros of f(x) are 1, –4, and 2

7. Find all real zeros of Find all possible rational zeros of: The possible rational zeros are Use synthetic division

8. Example Continued • This new factor has the same possible rational zeros: • Check to see if -1 is also a zero of this: • Conclusion:

9. Example Continued • This new factor has as possible rational zeros: • Check to see if -1 is also a zero of this: • Conclusion:

10. Example Continued • Check to see if 1 is a zero: • Conclusion:

11. Example Continued • Check to see if 2 is a zero: • Conclusion:

12. Example Continued • Summary of work done: is a zero of multiplicity two; 2 is a zero; and the other two zeros can be found by solving:

13. Using The Linear Factorization Theorem Find a 4th degree polynomial function with real coefficients that has as zeros and such that . Solution: Because is a zero , the conjugate, , must also be a zero. We can now use the Linear Factorization Theorem for a fourth-degree polynomial.

14. Using The Linear Factorization Theorem Substituting for in the formula for , we obtain

15. Descartes Rule of Signs is a method for determining the number of sign changes in a polynomial function.

16. Descarte’s Rule of Signs and Positive Real Zeros How do we determine the possible number of negative answers? We substitute for every x-value in the equation. Then we look for the sign changes.

18. 1 3 2 Descarte’s Rule of Signs Example Determine the possible number of positive real zeros and negative real zeros of P(x) = x4 – 6x3 + 8x2 + 2x – 1. We first consider the possible number of positive zeros by observing that P(x) has three variations in signs. +x4 – 6x3 + 8x2 + 2x – 1 Thus, by Descartes’ rule of signs, f has either 3 or 3 – 2 = 1 positive real zeros. For negative zeros, consider the variations in signs for P(x). P(x) = (x)4 – 6(x)3 + 8(x)2 + 2(x)  1 = x4 + 6x3 + 8x2 – 2x – 1 Since there is only one variation in sign, P(x) has only one negative real root. Total number of zeros 4 Positive: 3 1 Negative: 1 1 Nonreal: 0 2