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)

TEKS FOCUSConjugates – 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.

VOCABULARY

The factors of the numbers an and a0 in P (x) = anx n + an-1x n-1 + c + a1x + a0 can help you factor P (x) and solve the equation P (x) = 0.

ESSENTIAL UNDERSTANDING

Let P (x) = anx n + an-1x n-1 + c + a1x + 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.Rational roots must have reduced form

pq where p is an integer factor of a0 and q is

an integer factor of an.

21x2 + 29x + 10 = 0

x2 + 2921x + 10

21 = 0

(x + 23 )(x + 5

7 ) = 0

Theorem Rational Root Theorem

Factors of the leading coefficient:�1, �3, �7, and �21.

Factors of the constant term: �1, �2, �5, and �10.

The roots are �23 and �5

7.

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.

Theorem Conjugate Root Theorem

8-6 Theorems About Roots of Polynomial Equations

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Problem 2bl 2

Problem 1P

Finding a Rational Root

What are the rational roots of 2x 3 − x 2 + 2x + 5 = 0?

The only possible rational roots have the form factor of constant termfactor of leading coefficient.

The constant factors are {1, {5. The leading coefficient factors are {1, {2.

The only possible rational roots are {1, {5, { 12, {52.

The table shows the values of the function y = P (x) for the possible roots.

The only rational root of 2x 3 - x 2 + 2x + 5 = 0 is -1.

x

P(x)

52�

752�35

52

72

12�

6

12�5

�280

5

240

�1

0

1

8

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.

Theorem Descartes’ Rule of Signs

Using the Rational Root Theorem

What are the rational roots of 15x 3 − 32x 2 + 3x + 2 = 0?

Step 1 The constant term factors are {1 and {2. The leading coefficient factors are {1, {3, {5, and {15.

TEKS Process Standard (1)(E)

Coefficients and the constant term of the polynomial

The roots of the polynomial equation

continued on next page ▶

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TT

T

T

What information can you get from the equation?

370 Lesson 8-6 Theorems About Roots of Polynomial Equations

Problem 4bl

Problem 3bl 3

continuedProblem 2

Step 2 The possible rational roots are: {1, {2, {13, {2

3, {15, {2

5, { 115, and { 2

15.

Step 3 Test each possible rational root in 15x 3 - 32x 2 + 3x + 2 until you find a root.

Test 1: 15(1) 3 - 32(1) 2 + 3(1) + 2 = -12 ≠ 0

Test 2: 15(2) 3 - 32(2) 2 + 3(2) + 2 = 0

Step 4 Factor the polynomial by using synthetic division: P (x) = (x - 2)115x 2 - 2x - 12.

Step 5 Since 15x 2 - 2x - 1 = (5x + 1)(3x - 1), the other roots are -15 and 13.

The rational roots of 15x 3 - 32x 2 + 3x + 2 = 0 are 2, -15, and 13.

15 -32 3 2

30 -4 -2

15 -2 -1 0

2

Using the Conjugate Root Theorem to Identify Roots

A quartic polynomial P (x) has rational coefficients. If 12 and 1 + i are roots of P (x) = 0, what are the two other roots?

Since P (x) has rational coefficients and 0 + 12 is a root of P (x) = 0, it follows from the Conjugate Root Theorem that 0 - 12 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 -12 and 1 - i.

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 2i?

P (x) = x 3 - 2x 2 - 16x + 32 P (x) = x 3 + 4x 2 + 4x + 16

P (x) = x 3 - 4x 2 + 4x - 16 P (x) = x 3 + 4x 2 - 4x - 16

Since 2i is a root, then -2i is also a root.

P (x) = (x + 2i)(x - 2i)(x + 4)

= (x 2 + 4)(x + 4)

= x 3 + 4x 2 + 4x + 16

The equation x 3 + 4x 2 + 4x + 16 = 0 has rational coefficients and has roots -4 and 2i. The correct answer is C.

S

Does the Conjugate Root Theorem apply to −4?No; the theorem does not apply because -4is neither irrational nor

AP

SSt

Sa

Do you have real coefficients?All rational numbers are

the rational coefficients

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Problem 5P

Using Descartes’ Rule of Signs

What does Descartes’ Rule of Signs tell you about the real roots of x 3 − x 2 + 1 = 0?

There are two sign changes, + to - and - to + .Therefore, there are either 0 or 2 positive real roots.

P (-x) = (-x)3 - (-x)2 + 1 = -x 3 - x 2 + 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.

TEKS Process Standard (1)(F)

PRACTICE and APPLICATION EXERCISES

ONLINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots.

1. x 3 - 4x + 1 = 0 2. x 3 + 2x - 9 = 0 3. 2x 3 - 5x + 4 = 0

4. 3x 3 + 9x - 6 = 0 5. 4x 3 + 2x - 12 = 0 6. 6x 3 + 2x - 18 = 0

7. 7x 3 - x 2 + 4x + 10 = 0 8. 8x 3 + 2x 2 - 5x + 1 = 0 9. 10x 3 - 7x 2 + x - 10 = 0

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 cm3 of clay, what is the greatest height you could use for your pyramid?

Scan page for a Virtual Nerd™ tutorial video.

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TT

PT

Rg

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.

372 Lesson 8-6 Theorems About Roots of Polynomial Equations

A polynomial function P (x) with rational coefficients has the given roots. Find two additional roots of P (x) = 0.

11. -2i and 110 12. 14 - 12 and -6i

13. i and 7 + 8i 14. -13 and 5 - 111

Write a polynomial function with rational coefficients so that P (x) = 0 has the given roots.

15. 7 and 12 16. -9 and -15

17. -10i 18. 3i + 9

19. 4, 16, and 1 + 19i 20. 13i and 5 + 10i

21. 11 - 2i and 8 + 13i 22. 17 - 4i and 12 + 5i

What does Descartes’ Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function?

23. P (x) = x 2 + 5x + 6 24. P (x) = 9x 3 - 4x 2 + 10 25. P (x) = 8x 3 + 2x 2 - 14x + 5

26. Evaluate Reasonableness (1)(B) Your friend is using Descartes’ Rule of Signs to find the number of negative real roots of x 3 + x 2 + x + 1 = 0. Describe and correct the error.

27. 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.

28. 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 ft2 bouldering area, what dimensions should they use in the gym?

P(–x) = (–x)3 + (–x)2 + (–x) + 1 = –x3 – x2 – x + 1Because there is only one signchange in P(–x), there must be onenegative real root.

2h

h

2h + 4

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TEXAS Test PracticeT

39. What is a positive root of -5x3 - 2x2 + 9x + 30 = 0?

40. What is the remainder when you divide x3 + 2x2 - x - 6 by x - 1?

41. A polynomial with rational coefficients has roots -3i and 8 + 27. What is the minimum degree of the polynomial?

42. What is the value of y in the solution of the system of equations below?

{10x + 24y = 9

8x + 60y = 14

43. What is the value of the greater solution of the equation 6x2 - 17x + 5 = 0?

Find all rational roots for P (x) = 0.

29. P (x) = 2x 3 - 5x 2 + x - 1

30. P (x) = 6x 4 - 13x 3 + 13x 2 - 39x - 15

31. P (x) = 7x 3 - x 2 - 5x + 14

32. P (x) = 3x 4 - 7x 3 + 10x 2 - x + 12

33. P (x) = 6x 4 - 7x 2 - 3

34. P (x) = 2x 3 - 3x 2 - 8x + 12

35. 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 1 + 12 is the only root.

b. Find a polynomial equation with root 1 + 12 of multiplicity 2.

c. Find c such that 1 + 12 is a root of x 2 - 2x + c = 0.

37. 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 2 + 13 is a root of x 2 - 2x - (3 + 213) = 0. The student claims that 2 - 13 is another root of the equation by the Conjugate Root Theorem. Explain how you would respond to the student.

374 Lesson 8-6 Theorems About Roots of Polynomial Equations