Real Zeros of Polynomial Functions

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Presentation transcript:

Real Zeros of Polynomial Functions Section 2.3 Real Zeros of Polynomial Functions

Objective By following instructions students will be able to: Use long division to divide polynomials by other polynomials. Use synthetic division to divide polynomials by binomials of the form (x-k). Use the Remainder Theorem and the Factor Theorem. Use the Rational Zero Test to determine possible rational zeros of polynomial functions. Determine upper and lower bounds for zeros of polynomial functions.

Example 1: Divide by , and use the result to factor the function f completely.

Division Algorithm Ex.

Example 2: Divide using long division. a) by b) by

Example 3: Divide by .

U-TRY #1 Determine if the equations are equivalent by dividing. and

Synthetic Division A process used to find a factor of a polynomial. May only be used when dividing by a linear binomial.

Example 4: Use synthetic division to divide by .

Remainder Theorem If a polynomial f(x) is divided by x-k, the remainder is r=f(k).

Example 5: Divide using synthetic division. Use the Remainder Theorem to check your work. when

U-TRY #2 Divide using synthetic division. Use the remainder theorem to check your work. a) b)

Factor Theorem A polynomial f(x) has a factor (x-h) if and only if f(x)=0.

Example 6: Show that and are factors of

Rational Zero Test Possible rational zeros of a polynomial

Example 7: Find the rational zeros of .

Example 8: Find the rational zeros of .

Example 9: Find the real zeros of .

U-TRY #3 List all the possible rational zeros of the function. a) b)

Upper and Lower Bound Rules Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x-c, using synthetic division. 1. If c>0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c<0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f.

Example 10: Find the real zeros of .

Revisit Objective Did we… Use long division to divide polynomials by other polynomials? Use synthetic division to divide polynomials by binomials of the form (x-k)? Use the Remainder Theorem and the Factor Theorem? Use the Rational Zero Test to determine possible rational zeros of polynomial functions? Determine upper and lower bounds for zeros of polynomial functions?

Homework HW: I: pg 131 #8 pg 157 #s 60,72, II: pg 170 #1, 7, 9, 13, 15, 21, 27, 35, 37, 39, 43, 47, 55, 63, 69, 70 III:pg 173 #s 93-95