5-8 Study Guide and Intervention

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1 Study Guide and Intervention Identify Rational Zeros Rational Zero Theorem Corollary (Integral Zero Theorem) Let f(x) = a n x n + a n - 1 x n a 2 x 2 + a 1 x + a 0 represent a polynomial function with integral coefficients. If p q is a rational number in simplest form and is a zero of y = f(x), then p is a factor of a 0 and q is a factor of a n. If the coefficients of a polynomial are integers such that a n = 1 and a 0 0, any rational zeros of the function must be factors of a 0. Example a. f(x) = 3x - 2x 2 + 6x - 10 If p q is a rational root, then p is a factor of -10 and q is a factor of 3. The possible values for p are ±1, ±2, ±5, and ±10. The possible values for q are 61 and 63. So all of the possible rational zeros are p q = ±1, ±2, ±5, ±10, ± 1 3, ± 2 3, ± 5 3, and ± b. q(x) = x 3-10x 2 + 1x - 36 Since the coefficient of x 3 is 1, the possible rational zeros must be the factors of the constant term -36. So the possible rational zeros are ±1, ±2, ±3, ±, ±6, ±9, ±12, ±18, and ± f(x) = x 3 + 3x 2 - x g(x) = x 5-7x + 3x 2 + x h(x) = x - 7x 3 - x 2 + x - 9. p(x) = 2x - 5x 3 + 8x 2 + 3x q(x) = 3x - 5x x r(x) = x 5-2x f(x) = x 7-6x 5-3x + x 3 + x g(x) = 5x 6-3x + 5x 3 + 2x h(x) = 6x 5-3x + 12x x 2-9x p(x) = 2x 7-3x x 5-20x Lesson Chapter 5 7 Glencoe Algebra 2

2 Find Rational Zeros Example 1 Study Guide and Intervention (continued) Find all of the rational zeros of f(x) = 5x x 2-29x From the corollary to the Fundamental Theorem of Algebra, we know that there are exactly 3 complex roots. According to Descartes Rule of Signs there are 2 or 0 positive real roots and 1 negative real root. The possible rational zeros are ±1, ±2, ±3, ±, ±6, ±12, ± 1 5, ± 2 5, ± 3 5, ± 5, ± 6 5, ± 12. Make a table and test some possible rational zeros. 5 p q Since f(1) = 0, you know that x = 1 is a zero. The depressed polynomial is 5x x - 12, which can be factored as (5x - 3)(x + ). By the Zero Product Property, this expression equals 0 when x = 3 or x = -. 5 The rational zeros of this function are 1, 3, and -. 5 Example 2 Find all of the zeros of f(x) = 8x + 2x 3 + 5x 2 + 2x - 3. There are complex roots, with 1 positive real root and 3 or 1 negative real roots. The possible rational zeros are ±1, ±3, ± 1 2, ± 1, ± 1 8, ± 3 2, ± 3, and ± 3 8. Make a table and test some possible values. p q = 0, we know that x = 2) 1 2 The depressed polynomial is 8x 3 + 6x 2 + 8x + 6. Try synthetic substitution again. Any remaining rational roots must be negative. Since f ( 1 x = - 3 is another rational root. is a root. The depressed polynomial is 8x = 0, which has roots ±i. The zeros of this function are 1, - 3, and ±i Find all of the rational zeros of each function. 1. f(x) = x 3 + x 2-25x f(x) = x 3 + 6x 2 + x + 2 p q Find all of the zeros of each function. 3. f(x) = x + 2x 3-11x 2 + 8x f(x) = x + 5x x 2 + 5x - 5 Chapter 5 8 Glencoe Algebra 2

3 Skills Practice 1. n(x) = x 2 + 5x h(x) = x 2-2x w(x) = x 2-5x f(x) = 2x 2 + 5x q(x) = 6x 3 + x 2 - x g(x) = 9x + 3x 3 + 3x 2 - x + 27 Find all of the rational zeros of each function. 7. f(x) = x 3-2x 2 + 5x - 8. g(x) = x 3-3x 2 - x p(x) = x 3 - x 2 + x z(x) = x 3 - x 2 + 6x h(x) = x 3 - x 2 + x g(x) = 3x 3-9x 2-10x - 8 Lesson 13. g(x) = 2x 3 + 7x 2-7x h(x) = 2x 3-5x 2 - x p(x) = 3x 3-5x 2-1x q(x) = 3x 3 + 2x x q(x) = 3x 3-7x f(x) = x - 2x 3-13x 2 + 1x p(x) = x - 5x 3-9x 2-25x n(x) = 16x - 32x 3-13x x - 6 Find all of the zeros of each function. 21. f(x) = x 3 + 5x x q(x) = x 3-10x x m(x) = 6x - 17x 3 + 8x 2 + 8x g(x) = x + x 3 + 5x 2 + x + Chapter 5 9 Glencoe Algebra 2

4 Practice 1. h(x) = x 3-5x 2 + 2x s(x) = x - 8x 3 + 7x f(x) = 3x 5-5x 2 + x + 6. p(x) = 3x 2 + x g(x) = 5x 3 + x 2 - x q(x) = 6x 5 + x 3-3 Find all of the rational zeros of each function. 7. q(x) = x 3 + 3x 2-6x v(x) = x 3-9x x c(x) = x 3 - x 2-8x f(x) = x - 9x h(x) = x 3-7x x b(x) = x 3 + 6x f(x) = x 3-6x 2 + x g(x) = 2x 3 + 3x 2 - x h(x) = 2x 3-7x 2-21x z(x) = x - 3x 3 + 5x 2-27x d(x) = x + x n(x) = x - 2x p(x) = 2x - 7x 3 + x 2 + 7x q(x) = 6x + -9x 3 + 0x 2 + 7x - 12 Find all of the zeros of each function. 21. f(x) = 2x + 7x 3-2x 2-19x q(x) = x - x 3 + x x h(x) = x 6-8x 3 2. g(x) = x TRAVEL The height of a box that Joan is shipping is 3 inches less than the width of the box. The length is 2 inches more than twice the width. The volume of the box is 150 in 3. What are the dimensions of the box? 26. GEOMETRY The height of a square pyramid is 3 meters shorter than the side of its base. If the volume of the pyramid is 32 m 3, how tall is it? Use the formula V = 1 3 Bh. Chapter 5 50 Glencoe Algebra 2

5 Word Problem Practice 1. ROOTS Paul was examining an old algebra book. He came upon a page about polynomial equations and saw the polynomial below. x As you can see, all the middle terms were blotted out by an ink spill. What are all the possible rational roots of this polynomial?. PYRAMIDS The Great Pyramid in Giza, Egypt has a square base with side lengths of 5x yards and a height of x - 50 yards. The volume of the Great Pyramid is 3,125,000 cubic yards. Use a calculator to find the value of x and the dimensions of the pyramid. 5. BOXES Devon made a box with length x + 1, width x + 3, and height x IRRATIONAL CONSTANTS Cherie was given a polynomial whose constant term was 2. Is it possible for this polynomial to have a rational root? If it is not, explain why not. If it is possible, give an example of such a polynomial with a rational root. 3. MARKOV CHAINS Tara is a mathematician who specializes in probability. In the course of her work, she needed to find the roots of the polynomial p(x) = 288x - 288x x 2-17x + 1. What are the roots of p(x)? x - 3 x + 3 x + 1 a. What is the volume of Devon s box as a function of x? b. What is x if the volume of the box is equal to 1001 cubic inches? c. What is x if the volume of the box is equal to cubic inches? Lesson Chapter 5 51 Glencoe Algebra 2

6 Enrichment Irrational Numbers Philosopher Hippasus of Metapontum was believed to have discovered that 2 was irrational. Mathematicians of the time denied the existence of irrational numbers and killed Hippasus, not wishing to believe this fundamental number could fail to be a ratio of integers. The typical way to prove that 2 is irrational is by contradiction and relies on a few other common facts that are easily proven. That is, the proof assumes that it is rational and deduces a contradiction. Theorem: 2 is irrational Proof: Suppose 2 is a rational number. Then 2 = a, where a and b are relatively b prime integers. Relatively prime integers are integers that have no common factor other than one, therefore a is a fraction written in lowest terms. It is also this condition that b provides the contradiction. If we square both sides of the equation, 2 = a, we have b 2 = a 2 b. This is equivalent to 2 a2 = 2b 2. However, this says that a 2 is an even number, thus a is an even number. If a is even and a2 2 = b2, b is also even. Thus a and b have a factor in common other than one, namely two, and are not relatively prime. Hence 2 is irrational. The provides a direct proof method. 1. Use the rational zero theorem to prove that 2 is irrational. 2. Show that the square of an even number is even. 3. Show that any integer zeros of a polynomial function must be factors of the constant term a 0. Chapter 5 52 Glencoe Algebra 2

7 NAME DATE PERIOD Graphing Calculator Activity Rational Root Theorem The following program performs synthetic division and displays the depressed polynomial coefficients in rational form. The program will allow the testing of possible rational zeros of a polynomial function. PROGRAM: SYNTHDIV Disp DEGREE OF DIVIDEND P+1 P Q L 2 (P) Input M Disp COEFFICIENT P+1 P Disp COEFFICIENTS? Input A If P M+1 Disp 0=SAME A L 1 (P) Goto 3 Disp 1=QUOTIENT If P<M+1 Stop Disp 2=NEW Goto 1 Lbl Input U Lbl 2 0 P Disp POSSIBLE ROOT 1 P Lbl 5 Input R 0 S 1+P P If U=0 Lbl 3 L2(P) L1(P) Goto 2 L 1 (P) F If P<M+1 If U=1 F+S Q Goto 5 Goto Disp Q Frac Goto 2 0 P Pause Lbl 1 RQ S Example Find all of the rational zeros of f(x) = 2x 3-11x x + 9. Use the program to test possible zeros. Test the zero 9 first. Keystrokes: PRGM [SYNTHDIV] ENTER ENTER 3 ENTER 2 ENTER 1 ENTER 2 ENTER ( ) 11 ENTER 12 ENTER 9 ENTER. Press ENTER until the screen displays Done. The column of numbers are the coefficients of the depressed polynomial. Since the last number is not zero, 9 is not a solution. Test -1. Press ENTER 3 ENTER. Choose 0 for the same coefficients. Press ENTER ( ) 1 then ENTER until finished. Repeat this until a zero is found. Then press ENTER 2 for the degree of the depressed polynomial and ENTER 1 for the quotient. The zeros are 3, 3, and Find all the rational zeros of each function. 1. f(x) = x 3-8x 2-23x f(x) = x 3-7x 2 + 2x + 0 Lesson 3. f(x) = 2x 3 - x 2-32x f(x) = x + x 3-11x 2-9x p(x) = 3x + 11x x 2 + x p(x) = x - 2x 3 + x 2-8x p(x) = 3x 5 + x - 23x p(x) = 3x + 13x x 2 - Chapter 5 53 Glencoe Algebra 2