Theorems About Roots of Polynomial Equations. Rational Root Theorem

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1 8-6 Theorems About Roots of Polynomial Equations TEKS FOCUS TEKS (7)(E) Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping. TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. Additional TEKS (1)(F), (7)(A), (7)(D) VOCABULARY Conjugates number pairs of the form a + 1b and a - 1b Representation a way to display or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING The factors of the numbers a n and a 0 in P (x) = a n x n + a n-1 x n-1 + c + a 1 x + a 0 can help you factor P (x) and solve the equation P (x) = 0. Theorem Rational Root Theorem Let P (x) = a n x n + a n-1 x n-1 + c + a 1 x + a 0 be a polynomial with integer coefficients. There are a limited number of possible roots of P (x) = 0: Integer roots must be factors of a 0. Rational roots must have reduced form p q where p is an integer factor of a 0 and q is an integer factor of a n. Factors of the leading coefficient: 1, 3, 7, and 1. 1x + 9x + 10 = 0 x x = 0 ( x + 3)( x + 5 7) = 0 Factors of the constant term: 1,, 5, and 10. The roots are 3 and 5 7. Theorem Conjugate Root Theorem If P (x) is a polynomial with rational coefficients, then irrational roots of P (x) = 0 that have the form a + 1b occur in conjugate pairs. That is, if a + 1b is an irrational root with a and b rational, then a - 1b is also a root. If P (x) is a polynomial with real coefficients, then the complex roots of P (x) = 0 occur in conjugate pairs. That is, if a + bi is a complex root with a and b real, then a - bi is also a root. PearsonTEXAS.com 369

2 Theorem Descartes Rule of Signs Let P (x) be a polynomial with real coefficients written in standard form. The number of positive real roots of P (x) = 0 is either equal to the number of sign changes between consecutive coefficients of P (x) or is less than that by an even number. The number of negative real roots of P (x) = 0 is either equal to the number of sign changes between consecutive coefficients of P (-x) or is less than that by an even number. In both cases, count multiple roots according to their multiplicity. Problem 1 Finding a Rational Root What information can you get from the equation? What are the rational roots of x 3 x + x + 5 = 0? factor of constant term The only possible rational roots have the form factor of leading coefficient. The constant factors are {1, {5. The leading coefficient factors are {1, {. The only possible rational roots are {1, {5, { 1, {5. The table shows the values of the function y = P (x) for the possible roots. x P(x) The only rational root of x 3 - x + x + 5 = 0 is -1. Problem TEKS Process Standard (1)(E) Using the Rational Root Theorem What are the rational roots of 15x 3 3x + 3x + = 0? Coefficients and the constant term of the polynomial The roots of the polynomial equation Step 1 The constant term factors are {1 and {. The leading coefficient factors are {1, {3, {5, and {15. continued on next page 370 Lesson 8-6 Theorems About Roots of Polynomial Equations

3 Problem continued Step The possible rational roots are: {1, {, { 1 3, { 3, {1 5, { 5, { 1 15, and { 15. Step 3 Test each possible rational root in 15x 3-3x + 3x + until you find a root. Test 1: 15(1) 3-3(1) + 3(1) + =-1 0 Test : 15() 3-3() + 3() + = 0 Step 4 Factor the polynomial by using synthetic division: P (x) = (x - )115x - x Step 5 Since 15x - x - 1 = (5x + 1)(3x - 1), the other roots are and 1 3. The rational roots of 15x 3-3x + 3x + = 0 are, - 1 5, and 1 3. Problem 3 Using the Conjugate Root Theorem to Identify Roots Do you have real coefficients? All rational numbers are the rational coefficients A quartic polynomial P (x) has rational coefficients. If 1 and 1 + i are roots of P (x) = 0, what are the two other roots? Since P (x) has rational coefficients and is a root of P (x) = 0, it follows from the Conjugate Root Theorem that 0-1 is also a root. Since P (x) has real coefficients and 1 + i is a root of P (x) = 0, it follows that 1 - i is also a root. The two other roots are - 1 and 1 - i. Problem 4 Using Conjugates to Construct a Polynomial Multiple Choice What is a third-degree polynomial function y = P (x) with rational coefficients so that P (x) = 0 has roots 4 and i? Does the Conjugate Root Theorem apply to 4? No; the theorem does not apply because -4 is neither irrational nor P (x) = x 3 - x - 16x + 3 P (x) = x 3 + 4x + 4x + 16 P (x) = x 3-4x + 4x - 16 P (x) = x 3 + 4x - 4x - 16 Since i is a root, then -i is also a root. P (x) = (x + i)(x - i)(x + 4) = (x + 4)(x + 4) = x 3 + 4x + 4x + 16 The equation x 3 + 4x + 4x + 16 = 0 has rational coefficients and has roots -4 and i. The correct answer is C. PearsonTEXAS.com 371

4 Problem 5 TEKS Process Standard (1)(F) Using Descartes Rule of Signs Why can t there be zero negative real roots? The number of negative roots is equal to 1 or is less than 1 by an even number. Zero is less than 1 by an odd number. What does Descartes Rule of Signs tell you about the real roots of x 3 x + 1 = 0? There are two sign changes, + to - and - to +. Therefore, there are either 0 or positive real roots. P (-x) = (-x) 3 - (-x) + 1 =-x 3 - x + 1 = 0 has only one sign change, - to +. There is one negative real root. Recall that graphs of cubic functions have zero or two turning points. Because the graph already shows two turning points, it will not change direction again. So there are no positive real roots. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots. For additional support when completing your homework, go to PearsonTEXAS.com. 1. x 3-4x + 1 = 0. x 3 + x - 9 = 0 3. x 3-5x + 4 = x 3 + 9x - 6 = x 3 + x - 1 = x 3 + x - 18 = x 3 - x + 4x + 10 = x 3 + x - 5x + 1 = x 3-7x + x - 10 = Apply Mathematics (1)(A) You are building a square pyramid out of clay and want the height to be 0.5 cm shorter than twice the length of each side of the base. If you have 18 cm 3 of clay, what is the greatest height you could use for your pyramid? 37 Lesson 8-6 Theorems About Roots of Polynomial Equations

5 A polynomial function P (x) with rational coefficients has the given roots. Find two additional roots of P (x) = i and and -6i 13. i and 7 + 8i and Write a polynomial function with rational coefficients so that P (x) = 0 has the given roots and and i 18. 3i , 16, and i 0. 13i and i i and i. 17-4i and 1 + 5i What does Descartes Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function? 3. P (x) = x + 5x P (x) = 9x 3-4x P (x) = 8x 3 + x - 14x Evaluate Reasonableness (1)(B) Your friend is using Descartes Rule of Signs to find the number of negative real roots of x 3 + x + x + 1 = 0. Describe and correct the error. 7. Explain Mathematical Ideas (1)(G) A quartic equation with integer coefficients has two real roots and one imaginary root. Explain why the fourth root must be imaginary. P( x) = ( x) 3 + ( x) + ( x) + 1 = x 3 x x + 1 Because there is only one sign change in P( x), there must be one negative real root. 8. Apply Mathematics (1)(A) A climbing gym is designing a new bouldering section in the shape of a trapezoid. They want the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If they have enough shredded rubber to create a 60 ft bouldering area, what dimensions should they use in the gym? h h h + 4 PearsonTEXAS.com 373

6 Find all rational roots for P (x) = P (x) = x 3-5x + x P (x) = 6x 4-13x x - 39x P (x) = 7x 3 - x - 5x P (x) = 3x 4-7x x - x P (x) = 6x 4-7x P (x) = x 3-3x - 8x Connect Mathematical Ideas (1)(F) Write a fourth-degree polynomial equation with integer coefficients that has two irrational roots and two imaginary roots. 36. a. Find a polynomial equation in which is the only root. b. Find a polynomial equation with root of multiplicity. c. Find c such that is a root of x - x + c = a. Using real and imaginary as types of roots, list all possible combinations of root type for a fourth-degree polynomial equation. b. Repeat the process for a fifth-degree polynomial equation. c. Make a conjecture about the number of real roots of an odd-degree polynomial equation. 38. Justify Mathematical Arguments (1)(G) A student states that + 13 is a root of x - x - (3 + 13) = 0. The student claims that - 13 is another root of the equation by the Conjugate Root Theorem. Explain how you would respond to the student. TEXAS Test Practice 39. What is a positive root of -5x 3 - x + 9x + 30 = 0? 40. What is the remainder when you divide x 3 + x - x - 6 by x - 1? 41. A polynomial with rational coefficients has roots -3i and What is the minimum degree of the polynomial? 4. What is the value of y in the solution of the system of equations below? 10x + 4y = 9 { 8x + 60y = What is the value of the greater solution of the equation 6x - 17x + 5 = 0? 374 Lesson 8-6 Theorems About Roots of Polynomial Equations