Lesson 1: Solutions to Polynomial Equations

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

Classwork

Opening Exercise How many solutions are there to the equation 푥2 = 1? Explain how you know.

Example 1: Prove that a Quadratic Equation Has Only Two Solutions over the Set of Complex Numbers

Prove that 1 and −1 are the only solutions to the equation 푥2 = 1. Let 푥 = 푎 + 푏푖 be a complex number so that 푥2 = 1. a. Substitute 푎 + 푏푖 for 푥 in the equation 푥2 = 1.

b. Rewrite both sides in standard form for a complex number.

c. Equate the real parts on each side of the equation, and equate the imaginary parts on each side of the equation.

d. Solve for 푎 and 푏, and find the solutions for 푥 = 푎 + 푏푖.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

Exercises Find the product. 1. (푧 − 2)(푧 + 2)

2. (푧 + 3푖)(푧 − 3푖)

Write each of the following quadratic expressions as the product of two linear factors. 3. 푧2 − 4

4. 푧2 + 4

5. 푧2 − 4푖

6. 푧2 + 4푖

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

7. Can a quadratic polynomial equation with real coefficients have one real solution and one complex solution? If so, give an example of such an equation. If not, explain why not.

Recall from Algebra II that every quadratic expression can be written as a product of two linear factors, that is,

2 푎푥 + 푏푥 + 푐 = 푎(푥 − 푟1)(푥 − 푟2), 2 where 푟1 and 푟2 are solutions of the polynomial equation 푎푥 + 푏푥 + 푐 = 0.

8. Solve each equation by factoring, and state the solutions. a. 푥2 + 25 = 0

b. 푥2 + 10푥 + 25 = 0

9. Give an example of a quadratic equation with 2 + 3푖 as one of its solutions.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

10. A quadratic polynomial equation with real coefficients has a complex solution of the form 푎 + 푏푖 with 푏 ≠ 0. What must its other solution be and why?

11. Write the left side of each equation as a product of linear factors, and state the solutions. a. 푥3 − 1 = 0

b. 푥3 + 8 = 0

c. 푥4 + 7푥2 + 10 = 0

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

12. Consider the polynomial 푝(푥) = 푥3 + 4푥2 + 6푥 − 36. a. Graph 푦 = 푥3 + 4푥2 + 6푥 − 36, and find the real zero of polynomial 푝.

b. Write 푝 as a product of linear factors.

c. What are the solutions to the equation 푝(푥) = 0?

13. Malaya was told that the volume of a box that is a cube is 4,096 cubic inches. She knows the formula for the volume of a cube with side length 푥 is 푉(푥) = 푥3, so she models the volume of the box with the equation 푥3 − 4096 = 0. a. Solve this equation for 푥.

b. Malaya shows her work to Tiffany and tells her that she has found three different values for the side length of the box. Tiffany looks over Malaya’s work and sees that it is correct but explains to her that there is only one valid answer. Help Tiffany explain which answer is valid and why.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

14. Consider the polynomial 푝(푥) = 푥6 − 2푥5 + 7푥4 − 10푥3 + 14푥2 − 8푥 + 8. a. Graph 푦 = 푥6 − 2푥5 + 7푥4 − 10푥3 + 14푥2 − 8푥 + 8, and state the number of real zeros of 푝.

b. Verify that 푖 is a zero of 푝.

c. Given that 푖 is a zero of 푝, state another zero of 푝.

d. Given that 2푖 and 1 + 푖 are also zeros of 푝, explain why polynomial 푝 cannot possibly have any real zeros.

e. What is the solution set to the equation 푝(푥) = 0?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

15. Think of an example of a sixth-degree polynomial equation that, when written in standard form, has integer coefficients, four real number solutions, and two imaginary number solutions. How can you be sure your equation will have integer coefficients?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

Lesson Summary

Relevant Vocabulary

POLYNOMIAL FUNCTION: Given a polynomial expression in one variable, a polynomial function in one variable is a function 푓: ℝ → ℝ such that for each real number 푥 in the domain, 푓(푥) is the value found by substituting the number 푥 into all instances of the variable symbol in the polynomial expression and evaluating. It can be shown that if a function 푓: ℝ → ℝ is a polynomial function, then there is some nonnegative integer 푛 and

collection of real numbers 푎0, 푎1, 푎2,… , 푎푛 with 푎푛 ≠ 0 such that the function satisfies the equation 푛 푛−1 푓(푥) = 푎푛푥 + 푎푛−1푥 + ⋯ + 푎1푥 + 푎0, for every real number 푥 in the domain, which is called the standard form of the polynomial function. The function 푓(푥) = 3푥3 + 4푥2 + 4푥 + 7, where 푥 can be any real number, is an example of a function written in standard form.

DEGREE OF A POLYNOMIAL FUNCTION: The degree of a polynomial function is the degree of the polynomial expression used to define the polynomial function. The degree is the highest degree of its terms. The degree of 푓(푥) = 8푥3 + 4푥2 + 7푥 + 6 is 3, but the degree of 푔(푥) = (푥 + 1)2 − (푥 − 1)2 is 1 because when 푔 is put into standard form, it is 푔(푥) = 4푥.

ZEROS OR ROOTS OF A FUNCTION: A zero (or root) of a function 푓: ℝ → ℝ is a number 푥 of the domain such that 푓(푥) = 0. A zero of a function is an element in the solution set of the equation 푓(푥) = 0. Given any two polynomial functions 푝 and 푞, the solution set of the equation 푝(푥)푞(푥) = 0 can be quickly found by solving the two equations 푝(푥) = 0 and 푞(푥) = 0 and combining the solutions into one set. A number 푎 is zero of a polynomial function 푝 with multiplicity 푚 if the factored form of 푝 contains (푥 − 푎)푚. Every polynomial function of degree 푛, for 푛 ≥ 1, has 푛 zeros over the complex numbers, counted with multiplicity. Therefore, such polynomials can always be factored into 푛 linear factors.

Problem Set

1. Find all solutions to the following quadratic equations, and write each equation in factored form. a. 푥2 + 25 = 0 b. −푥2 − 16 = −7 c. (푥 + 2)2 + 1 = 0 d. (푥 + 2)2 = 푥 e. (푥2 + 1)2 + 2(푥2 + 1) − 8 = 0 f. (2푥 − 1)2 = (푥 + 1)2 − 3 g. 푥3 + 푥2 − 2푥 = 0 h. 푥3 − 2푥2 + 4푥 − 8 = 0

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS

2. The following cubic equations all have at least one real solution. Find the remaining solutions. a. 푥3 − 2푥2 − 5푥 + 6 = 0 b. 푥3 − 4푥2 + 6푥 − 4 = 0 c. 푥3 + 푥2 + 9푥 + 9 = 0 d. 푥3 + 4푥 = 0 e. 푥3 + 푥2 + 2푥 + 2 = 0

3. Find the solutions of the following equations. a. 4푥4 − 푥2 − 18 = 0 b. 푥3 − 8 = 0 c. 8푥3 − 27 = 0 d. 푥4 − 1 = 0 e. 81푥4 − 64 = 0 f. 20푥4 + 121푥2 − 25 = 0 g. 64푥3 + 27 = 0 h. 푥3 + 125 = 0

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