SECTION 3.5

1. SECTION 3.5 • REAL ZEROS OF A POLYNOMIAL FUNCTION

2. THE REAL ZEROS OF A POLYNOMIAL FUNCTION When we divide one polynomial by another, we obtain a quotient and a remainder. Thus the dividend can be written as: (Divisor)(Quotient) + Remainder

3. Dividend divisor quotient remainder THEOREM: DIVISION ALGORITHM FOR POLYNOMIALS OR f(x) = g(x) q(x) + r(x)

4. REAL ZEROS OF A POLYNOMIAL FUNCTION If the divisor is a polynomial of the form x - c where c is a real number, then the remainder r(x) is either the zero polynomial or a polynomial of degree 0. Thus, for such divisors, the remainder is some number R and we may write f(x) = (x - c) q(x) + R

5. REAL ZEROS OF A POLYNOMIAL FUNCTION If the x variable in the equation of f(x) gets replaced by the value c, then f(x) = (x - c) q(x) + R f(c) = (c - c) q(x) + R f(c) = R

6. REMAINDER THEOREM Let f be a polynomial function. If f(x) is divided by x - c, then the remainder is f(c). Ex: Find the remainder if f(x) = x3 - 4x2 + 2x - 5 is divided by (a) x - 3 and (b) x + 2

7. FACTOR THEOREM Let f be a polynomial function. Then x - c is a factor of f(x) if and only if f(c) = 0. Ex: Use the Factor Theorem to determine whether the function f(x) = 2x3 - x2 + 2x - 3 has the factor (a) x - 1 and (b) x + 3

8. THEOREM: NUMBER OF ZEROS A polynomial function cannot have more zeros than its degree.

9. DESCARTES’ RULE OF SIGNS Let f denote a polynomial function. The number of positive real zeros of f either equals the number of variations in sign of the nonzero coefficients of f(x) or else equals that number less an even integer.

10. DESCARTES’ RULE OF SIGNS Let f denote a polynomial function. The number of negative real zeros of f either equals the number of variations in sign of the nonzero coefficients of f(- x) or else equals that number less an even integer.

11. EXAMPLE Discuss the real zeros of f(x) = 3x6 - 4x4 + 3x3 + 2x2 - x - 3

12. RATIONAL ZEROS THEOREM Let f be a polynomial function of degree 1 or higher of the form f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 (an  0, a0  0) where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

13. EXAMPLE f(x) = 2x2 - x - 3 (2x - 3)(x + 1) zeros: 3/2, - 1

14. EXAMPLE List the potential rational zeros of f(x) = 2x3 + 11x2 - 7x - 6 p:  1,  2,  3,  6 q:  1,  2

15. FINDING THE REAL ZEROS OF A POLYNOMIAL FUNCTION EXAMPLES 5, 6 & 7

16. THEOREM Every polynomial function (with real coefficients) can be uniquely factored into a product of linear factors and/or irreducible quadratic factors.

17. COROLLARY Every polynomial function (with real coefficients) of odd degree has at least one real zero.

18. BOUNDS ON ZEROS We won’t worry about this topic.

19. Intermediate Value Theorem Let f denote a continuous function. If a < b and if f (a) and f (b) are of opposite signs, then the graph of f has at least one x-intercept between a and b.

20. EXAMPLE Use the Intermediate Value Theorem to show that the graph of the function has an x-intercept in the given interval. Approximate the x-intercept correct to 2 decimal places. f(x) = x4 + 8x3 - x2 + 2; [- 1, 0]

21. f(x) = x 4 + 8x 3 - x 2 + 2; [- 1, 0] f(-1) = (- 1)4 + 8(- 1)3 - (- 1)2 + 2 = 1 - 8 - 1 + 2 = - 6 f(0) = (0)4 + 8(0)3 - (0)2 + 2 = 0 + 0 - 0 + 2 = 2

22. EXAMPLE Use the IVT to show that the graph of the function has an x-intercept in the given interval. Approximate the x-intercept correct to 2 decimal places. f(x) = x 5 - 3x 4 - 2x 3 +6x 2 + x + 2; [1.7,1.8] Use your calculator for f(1.7) and f(1.8).

23. CONCLUSION OF SECTION 3.5