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May 24th

May 24th

# Real Zeros of Polynomial Functions

Objective:
In this lesson you learned how to use long division and
synthetic division to divide polynomials by other
polynomials and how to find the rational and real zeros of
polynomial
functions.

 Important Vocabulary Define each term or concept . Long division of polynomials A procedure for dividing two polynomials , which is similar to long division in arithmetic. Division Algorithm If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x) = d(x)q(x) + r(x) where r(x) = 0 or the degree of r(x) is less than the degree of d(x). Synthetic division A shortcut for long division of polynomials when dividing by divisors of the form x – k. Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is r = f(k). Factor Theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Upper bound A real number b is an upper bound for the real zeros of f if no real zeros of f are greater than b. Lower bound A real number b is a lower bound for the real zeros of f if no real zeros of f are less than b.
 I. Long Division of Polynomials (Pages 160-162) When dividing a polynomial f(x) by another polynomial d(x), if the remainder r(x) = 0, d(x) divides evenly into f(x). The rational ex pression f (x)/d(x) is improper if . . . the degree of f(x) is greater than or equal to the degree of d(x). The rational expression r (x)/d(x) is proper if . . . the degree of r(x) is less than the degree of d(x). The result of a division problem can be checked by . . . multiplying the quotient by the divisor and adding any remainder to see if the original dividend is obtained. Example 1: Divide 3x3 + 4x - 2 by x2 + 2x +1 . 3x - 6 + (13x + 4)/(x2 + 2x + 1) What you should learn How to use long division to divide polynomials by other polynomials II. Synthetic Division (Page 163) Can synthetic division be used to divide a polynomial by x2 - 5? Explain. No, the divisor must be in the form x - k. Can synthetic division be used to divide a polynomial by x + 4? Explain. Yes, rewrite x + 4 as x - (- 4). Example 2: Fill in the following synthetic division array to divide 2x4 + 5x2 - 3 by x - 5. Then carry out the synthetic division and indicate which entry represents the remainder. What you should learn How to use synthetic division to divide polynomials by binomials of the form (x - k) III. The Remainder and Factor Theorems (Pages 164-165) To use the Remainder Theorem to evaluate a polynomial function f(x) at x = k, . . . use synthetic division to divide f(x) by x - k. The remainder will be f(k). Example 3: Use the Remainder Theorem to evaluate the function f (x) = 2x4 + 5x2 - 3 at x = 5. 1372To use the Factor Theorem to show that (x - k) is a factor of a polynomial function f(x), . . . use synthetic division on f(x) with the factor (x - k). If the remainder is 0, then (x - k) is a factor. Or, alternatively, evaluate f(x) at x = k. If the result is 0, then (x - k) is a factor. List three facts about the remainder r, obtained in the synthetic division of f(x) by x - k: 1) The remainder r gives the value of f at x = k. That is, r = f(k). 2) If r = 0, (x - k) is a factor of f(x). 3) If r = 0, (k, 0) is an x- intercept of the graph of f. What you should learn How to use the Remainder Theorem and the Factor Theorem IV. The Rational Zero Test (Pages 166-168) Describe the purpose of the Rational Zero Test. The Rational Zero Test relates the possible rational zeros of a polynomial with integer coefficients to the leading coefficient and to the constant term of the polynomial. State the Rational Zero Test. If the polynomial has integer coefficients, every rational zero of f has the form: rational zero = p/q, where p and q have no common factors other than 1, and p = a factor of the constant term a0, and q = a factor of the leading coefficient an. To use the Rational Zero Test, . . . first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Then use trial and error to determine which of these possible rational zeros, if any, are actual zeros of the polynomial. Example 4: List the possible rational zeros of the polynomial function ± 1, ± 5, ± 1/3, ± 5/3 Some strategies that can be used to shorten the search for actual zeros among a list of possible rational zeros include . . . using a programmable calculator to speed up the calculations, using a graphing utility to estimate the locations of zeros, or using the Factor Theorem and synthetic division to test possible rational zeros, etc. What you should learn How to use the Rational Zero Test to determine possible rational zeros of polynomial functions V. Bounds for Real Zeros of Polynomial Functions (Pages 168-169) State the 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. Explain how the Upper and Lower Bound Rules can be useful in the search for the real zeros of a polynomial function. Explanations will vary. For instance, suppose you are checking a list of possible rational zeros. When checking the possible rational zero 2 with synthetic division, each number in the last row is positive or zero. Then you need not check any of the other possible rational zeros that are greater than 2 and can concentrate on checking only values less than 2. Additional notes What you should learn How to determine upper and lower bounds for zeros of polynomial functions Homework As signment Page(s) Exercises
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