- Theorems About Roots of Polynomial Equations Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. Also N.CN.8 Objectives To solve equations using the Rational Root Theorem To use the Conjugate Root Theorem I am greater than my square. The sum of my numerator and denominator is. What fraction am I? How did you find me? My numerator is a factor of 6. My denominator is a factor of 4. Lesson Vocabulary Rational Root Theorem Conjugate Root Theorem Descartes Rule of Signs Factoring the polynomial P (x) anx n anx n c ax a0 can be challenging, especially when both an and a0 have many factors. Essential Understanding The factors of the numbers an and a0 in P (x) anx n anx n c ax a0 can help you factor P (x) and solve the equation P (x) 0. One way to find a root of the polynomial equation P (x) 0 is to guess and check. This is inefficient unless there is a way to minimize the number of guesses, or possible roots. The Rational Root Theorem does just that. Theorem Rational Root Theorem Let P (x) anx n anx n c ax a0 be a polynomial with integer coefficients. There are a limited number of possible roots of P (x) 0: Integer roots must be factors of a0. p Rational roots must have reduced form q where p is an integer factor of a0 and q is an integer factor of an. Factors of the leading coefficient:, 3, 7, and. x 9x 0 0 x 9 0 x 0 Q x 3 RQ x 7 R 0 3 Chapter 03_hsmase_cc_00.indd 3 Factors of the constant term:,,, and 0. The roots are 3 and 7. Polynomials and Polynomial Functions 3/9/ :9:7 AM
Problem Finding a Rational Root What information can you get from the equation? The equation gives you the leading coefficient and the constant term. What are the rational roots of x 3 x x 0? factor of constant term The only possible rational roots have the form factor of leading coefficient. The constant factors are 4, 4. The leading coefficient factors are 4, 4. The only possible rational roots are 4, 4, 4, 4. The table shows the values of the function y P (x) for the possible roots. x P(x) 8 0 40 80 6 7 3 7 The only rational root of x 3 x x 0 is.. What are the rational roots of 3x 3 7x 6x 8 0? Once you find one root, use synthetic division to factor the polynomial. Continue finding roots and dividing until you have a second-degree polynomial. Use the Quadratic Formula to find the remaining roots. Problem Using the Rational Root Theorem What are the rational roots of x 3 3x 3x 0? Coefficients and the constant term of the polynomial The roots of the polynomial equation Find one root. Factor until you get a quadratic. Use the Quadratic Formula to find the other roots. Step The constant term factors are 4 and 4. The leading coefficient factors are 4, 43, 4, and 4. Step The possible rational roots are: 4, 4, 4 3, 4 3, 4, 4, 4, and 4. Step 3 Step 4 Test each possible rational root in x 3 3x 3x until you find a root. Test : () 3 3() 3() Z 0 Test : () 3 3() 3() 0 So is a root. Factor the polynomial by using synthetic division: P (x) (x )Ax x B. Step Since x x (x )(3x ), the other roots are and 3. The rational roots of x 3 3x 3x 0 are,, and 3. 3 3 30 4 0. What are the rational roots of x 3 x 7x 6 0? Lesson - Theorems About Roots of Polynomial Equations 33 mase_cc_00.indd 33 3/9/ :
Recall from Lesson 4-8 that the complex numbers a bi and a bi are conjugates. Similarly, the irrational numbers a!b and a!b are conjugates. If a complex number or an irrational number is a root of a polynomial equation with rational coefficients, so is its conjugate. 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!b occur in conjugate pairs. That is, if a!b is an irrational root with a and b rational, then a!b 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. Do you have real coefficients? All rational numbers are real numbers. Therefore the rational coefficients are real coefficients. Problem 3 Using the Conjugate Root Theorem to Identify Roots A quartic polynomial P (x) has rational coefficients. If! and i are roots of P (x) 0, what are the two other roots? Since P (x) has rational coefficients and 0! is a root of P (x) 0, it follows from the Conjugate Root Theorem that 0! is also a root. Since P (x) has real coefficients and i is a root of P (x) 0, it follows that i is also a root. The two other roots are! and i. 3. A cubic polynomial P (x) has real coefficients. If 3 i and are two roots of P (x) 0, what is one additional root? Does the Conjugate Root Theorem apply to 4? No; the theorem does not apply because 4 is neither irrational nor imaginary. 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 roots4 and i? P (x) x 3 x 6x 3 P (x) x 3 4x 4x 6 P (x) x 3 4x 4x 6 P (x) x 3 4x 4x 6 Since i is a root, then i is also a root. P (x) (x i)(x i)(x 4) Write the polynomial function. (x 4)(x 4) Multiply the complex conjugates. x 3 4x 4x 6 Write the polynomial function in standard form. The equation x 3 4x 4x 6 0 has rational coefficients and has roots 4 and i. The correct answer is C. 4. What quartic polynomial equation has roots 3i, 8,? 34 Chapter Polynomials and Polynomial Functions mase_cc_00.indd 34 3/9/ :
The French mathematician René Descartes (96 60) recognized a connection between the roots of a polynomial equation and the and signs of the standard form. 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. Why can t there be zero negative real roots? The number of negative roots is equal to or is less than by an even number. Zero is less than by an odd number. Problem Using Descartes Rule of Signs What does Descartes Rule of Signs tell you about the real roots of x 3 x 0? There are two sign changes, to and to. Therefore, there are either 0 or positive real roots. P (x) (x) 3 (x) x 3 x 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.. a. What does Descartes Rule of Signs tell you about the real roots of x 4 x 3 3x 0? b. Reasoning Can you confirm real and complex roots graphically? Explain. Lesson Check Do you know HOW? Use the Rational Root Theorem to list all possible rational roots for each equation.. x x 0. x 3 x 6 0 3. 3x 4 x 0 Write a polynomial function with rational coefficients so that P(x) 0 has the given roots. 4. and 9. 4 and i Do you UNDERSTAND? MATHEMATICAL PRACTICES 6. Vocabulary Give an example of a conjugate pair. 7. Reasoning In the statements below, r and s represent integers. Is each statement always, sometimes, or never true? Explain. a. A root of the equation 3x 3 rx sx 8 0 could be. b. A root of the equation 3x 3 rx sx 8 0 could be. 8. Error Analysis A student claims that 4i is the only imaginary root of a polynomial equation that has real coefficients. What is the student s mistake? Lesson - Theorems About Roots of Polynomial Equations 3 mase_cc_00.indd 3 3/9/ :
Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots. See Problems and. 9. x 3 4x 0 0. x 3 x 9 0. x 3 x 4 0. 3x 3 9x 6 0 3. 4x 3 x 0 4. 6x 3 x 8 0. 7x 3 x 4x 0 0 6. 8x 3 x x 0 7. 0x 3 7x x 0 0 A polynomial function P (x) with rational coefficients has the given roots. Find two additional roots of P (x) 0. See Problem 3. 8. i and!0 9. 4! and 6i 0. i and 7 8i.!3 and! Write a polynomial function with rational coefficients so that P (x) 0 has the given roots. See Problem 4.. 7 and 3. 9 and 4. 0i. 3i 9 6. 4, 6, and 9i 7. 3i and 0i 8. i and 8 3i 9. 7 4i and i What does Descartes Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function? See Problem. B Apply 30. P (x) x x 6 3. P (x) 9x 3 4x 0 3. P (x) 8x 3 x 4x Find all rational roots for P (x) 0. 33. P (x) x 3 x x 34. P (x) 6x 4 3x 3 3x 39x 3. P (x) 7x 3 x x 4 36. P (x) 3x 4 7x 3 0x x 37. P (x) 6x 4 7x 3 38. P (x) x 3 3x 8x Write a polynomial function P (x) with rational coefficients so that P (x) 0 has the given roots. 39. 6, 3, and i 40. 4! and 8i 4. 7i and! 4. Think About a Plan You are building a square pyramid out of clay and want the height to be 0. cm shorter than twice the length of each side of the base. If you have 8 cm 3 of clay, what is the greatest height you could use for your pyramid? How can drawing a diagram help you solve this problem? What is the formula for the volume of a pyramid? What equation can you solve to find the height of the pyramid? 43. Error Analysis Your friend is using Descartes Rule of Signs to find the number of negative real roots of x 3 x x 0. Describe and correct the error. 44. Reasoning A quartic equation with integer coefficients has two real roots and one imaginary root. Explain why the fourth root must be imaginary. 36 Chapter Polynomials and Polynomial Functions mase_cc_00.indd 36 3/9/ :
C Challenge 4. Gardening A gardener is designing a new garden in the shape of a trapezoid. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If she has enough topsoil to create a 60 ft garden, what dimensions should she use for the garden? 46. Open-Ended Write a fourth-degree polynomial equation with integer coefficients that has two irrational roots and two imaginary roots. 47. 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 0. 48. 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 Make a conjecture about the number of real roots of an odd-degree polynomial equation. 49. Writing A student states that!3 is a root of x x (3!3) 0. The student claims that!3 is another root of the equation by the Conjugate Root Theorem. Explain how you would respond to the student. Standardized Test Prep SAT/ACT 0. What is a positive root of x 3 x 9x 30 0?. What is the remainder when you divide x 3 x x 6 by x?. A polynomial with rational coefficients has roots 3i and 8!7. What is the minimum degree of the polynomial? 0x 4y 9 3. What is the value of y in the solution of the system of equations? b 8x 60y 4 4. What is the value of the greater solution of the equation 6x 7x 0? Mixed Review Divide.. (x 3 x 3) 4 (x ) 6. (8x 3 x 7) 4 (x 6) See Lesson -4. 7. (7x x 4) 4 (x ) Solve. 8. 7x 63 0 9. x 8 0 See Lesson 4-8. 60. x 88 0 Get Ready! To prepare for Lesson -6, do Exercises 6 and 6. Write each polynomial in standard form. Then classify it by degree and by number of terms. See Lesson -. 6. 6x x 4 9x 6. 3x 4x 7x 3 Lesson - Theorems About Roots of Polynomial Equations 37 mase_cc_00.indd 37 3/9/ :