1. Zeros of Polynomial Functions MATH 109 - Precalculus S. Rook

2. Overview Section 2.5 in the textbook: – Fundamental Theorem of Algebra – Rational Zeros Test – Complex Zeros 2

3. Fundamental Theorem of Algebra

4. Fundamental Theorem of Algebra: The polynomial function f(x) has EXACTLY n zeros where n is the degree of f(x) – The zeros may be: Real: – e.g. f(x) = x 2 – 1 = (x + 1)(x – 1) → x = -1, 1 (2 zeros) Imaginary – e.g. f(x) = x 2 + 1 = (x + i)(x – i) → x = -i, i (2 zeros) Complex – e.g. f(x) = x 2 – x + 1 → (2 zeros) 4

5. Fundamental Theorem of Algebra (Continued) A mixture of the two – e.g. f(x) = x 4 – 1 = (x 2 + 1)(x 2 – 1) = (x + i)(x – i)(x + 1)(x – 1) → x = -i, i, -1, 1 (4 zeros) – A zero x = a with multiplicity k contributes k zeros to the polynomial e.g. f(x) = x 3 – x 2 – x + 1 = (x – 1) 2 (x + 1) → x = -1, 1 [multiplicity 2] (3 zeros) 5

6. Fundamental Theorem of Algebra (Example) Ex 1: Given the following zeros, i) write each as a linear factor ii) combine the linear factors to obtain a polynomial f(x) with the given zeros x = 0, 2 + i, 2 – i 6

7. Rational Zeros Test

8. 8 Sometimes it is not always so simple to find the zeros of a polynomial – Especially when the degree goes beyond 2 Finding zeros is an important aspect of algebra Rational Zeros Test: If a polynomial is in the form f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0, then p / q where p is a factor of a 0 (constant term) and q is a factor of a n (leading term) represents ALL of the possible RATIONAL zeros of f(x) – i.e. List all the factors of p & q and write down all possible combinations of p / q being careful to exclude duplicates e.g. p = 1, q = 2  ± {1} / ± {1, 2}  ±{1, ½} f(x) must be in DESCENDING degree

9. Rational Zeros Test (Continued) To find the zeros of a polynomial function f(x) with a degree higher than 2: – Factor any possible GCF from the polynomial – Apply the Rational Zeros Test to list the potential RATIONAL zeros of f(x) – Use the potential zeros along with the Remainder Theorem to find a zero x = c such that f(c) = 0 Can be time-consuming because we have to use trial and error – Apply synthetic division using the zero x = c – Apply the appropriate strategy if the resulting polynomial has a degree of two or less; otherwise, repeat the Rational Zeros Test 9

10. Rational Zeros Test (Example) Ex 2: Find all the real zeros of the function: a)f(x) = x 3 + 2x 2 – x – 2 b)g(x) = x 4 – x 3 – 2x – 4 c)h(x) = x 3 – 6x 2 + 11x – 6 10

11. More on the Rational Zeros Test Bear in mind that the Rational Zeros Test only helps to find the RATIONAL zeros – i.e. CANNOT be used to find zeros that are: Irrational Imaginary – Apply the Rational Zeros Test to find all of the RATIONAL zeros Any irrational or imaginary solutions will come out in the end If no rational zeros exist, then we need to find another way to extract the zeros from the polynomial 11

12. More on the Rational Zeros Test (Example) Ex 3: Find all the real zeros of the function a)f(x) = x 3 – 2x 2 – x b)g(x) = x 3 + x 2 – 3x – 3 12

13. Complex Zeros

14. Conjugate Zeros Theorem: Let f(x) be a polynomial function with REAL coefficients. If x = a + bi is a zero of f(x), then so is its conjugate x = a – bi and vice versa – Recall that the product of a complex number and its conjugate produces a real number – Likewise, the product of a complex factor and its conjugate produces a factor with real coefficients e.g. (x – i)(x + i) = x 2 – i 2 = x 2 + 1 14

15. Complex Zeros (Continued) Given a complex zero of a polynomial function f(x): – Use synthetic division on f(x) with the given complex zero This will more than likely produce a function with complex coefficients – Use synthetic division on the result with the conjugate of the complex zero This will produce a polynomial with real coefficients – Use either the Rational Zeros Test or other strategies to break apart the polynomial 15

16. Complex Zeros (Example) Ex 4: Use the given zero to find all the zeros of the function: a)f(x) = 4x 3 + 23x 2 + 34x – 10;x = -3 + i b)g(x) = 2x 3 + 3x 2 + 50x + 75;x = 5i 16

17. Complex Zeros (Example) Ex 5: Find all the zeros of the function AND write the polynomial as a product of linear factors: a)f(x) = x 4 – 4x 3 + 8x 2 – 16x + 16 b)g(x) = x 3 – x + 6 17

18. Summary After studying these slides, you should be able to: – Understand the Fundamental Theorem of Algebra in terms of the number of zeros of a polynomial function – Write a list of zeros as linear factors and to combine a list of linear factors into a polynomial – Apply the Rational Zeros test to find the zeros of a polynomial Additional Practice – See the list of suggested problems for 2.5 Next lesson – Rational Functions (Section 2.6) 18