Download Theorems About Roots of Polynomial Equations

Theorems About Roots of
Polynomial Equations
5-5
Vocabulary
Review
1. Write L if the polynomial is linear, Q if it is quadratic, or C if it is cubic.
3x 1 x2
2x3 2 7x2 1 x
14 2 x
Vocabulary Builder
root (noun) root
Related Words: factor, solution, zero, x-intercept
Main Idea: A root is a solution of an equation. It is an x-intercept of the related
function, which is why it can be called a zero. If (x 2 a) is a factor of a polynomial,
then a is a root of that polynomial.
Use Your Vocabulary
Write T for true or F for false.
2. 1 and 21 are roots of the equation x2 5 1.
3. The equation x2 2 4x 1 4 5 0 has roots 4 and 24.
Write the number of roots each polynomial has.
4. 3x4 2 2x2 1 17x 2 4
Theorem
5. 12x5 1 x7 2 8 1 4x2
6. 15 1 6x
Rational Root Theorem
Let P(x) 5 anxn 1 an21xn21 1 c 1 a1x 1 a0 be a polynomial with integer
coefficients. Then there are a limited number of possible roots of P(x) 5 0.
Integer roots must be factors of a0 .
p
Rational roots must have a reduced form q , where p is an integer factor of a0 and
q is an integer factor of an .
Chapter 5
134
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Definition: A root of an equation is a value that, when substituted for the unknown
quantity, satisfies the equation.
What are the possible rational roots of 15x4 2 3x 1 8?
7. Identify a0 and an .
a0 5
an 5
9. List the factors of the leading coefficient, an .
8. List the factors of the constant, a0 .
4
,4
,4
,4
4
,
,4
,4
,4
,
10. Circle the possible rational roots.
5
4
2
5
215
2
3
Problem 2 Using the Rational Root Theorem
Got It? What are the rational roots of 2x3 1 x2 2 7x 2 6 5 0?
Underline the correct number(s) to complete each sentence.
11. The leading coefficient is 0 / 2 / 1 / 7 / 26 .
12. The constant is 0 / 2 / 1 / 7 / 26 .
13. The factors of the leading coefficient are 0 / 21 / 1 / 22 / 2 / 23 / 3 / 26 / 6 .
14. The factors of the constant are 0 / 21 / 1 / 22 / 2 / 23 / 3 / 26 / 6 .
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15. Cross out the numbers that are NOT possible rational roots.
1
3
21
0
1
2
16. Try the easiest possible roots.
P(1) 5
P(21) 5
17. Since 1 / 21 is a root, x 1 1 / x 2 1 is a factor of 2x3 1 x2 2 7x 2 6 5 0.
18. Use synthetic division to find another factor.
21
2
1
27
19. The quotient is
.
26
2
20. Use the quadratic formula to find the remaining factors.
2b 4 "b2 2 4ac
5
2a
5
4
4
"
2 4Q
5
RQ
R
5
4
"
2
or
21. The rational roots of 2x3 1 x2 2 7x 2 6 5 0 are
135
,
, and
.
Lesson 5-5
Theorem
Conjugate Root Theorem
If P(x) is a polynomial with rational coefficients, then the irrational roots of P(x) 5 0
occur in conjugate pairs. That is, if a 1 !b is an irrational root with a and b rational,
then a 2 !b is also a root.
If P(x) is a polynomial with real coefficients, then the complex roots of P(x) 5 0 occur
in conjugate pairs. That is, if a 1 bi is a complex root with a and b real, then a 2 bi is
also a root.
22. Write the conjugate of each root.
1 1 3i
Problem 3
4 2 !7
22 2 9i
15 1 !10
Using the Conjugate Root Theorem to Identify Roots
Got It? A cubic polynomial P(x)has real coefficients. If 3 2 2i and 52 are two roots
of P(x) 5 0, what is one additional root?
23. Place a ✓ if the Conjugate Root Theorem could be applied to the following types
of roots. Place an ✗ if it could not be.
Rational
Irrational
Complex
24. Write R if the root is rational, I if it is irrational, or C if it is complex.
25. By the Conjugate Root Theorem,
Problem 4
is an additional root.
Using Conjugates to Construct a Polynomial
Got It? What is a quartic polynomial function with rational coefficients for the
roots 2 2 3i, 8, 2?
Underline the correct word or number to complete each sentence.
26. A quartic polynomial has 1 / 2 / 3 / 4 roots.
27. Since 2 2 3i is a root,
is also a root.
28. Write P(x) as the product of four binomials.
P(x) 5 Q x
(2 2 3i) RQ x
8 RQ x
2 RQ x
R
29. Circle the simplified form of the polynomial.
x 4 2 14x3 1 37x2 2 66x 2 208
x 4 2 14x3 1 69x2 2 194x 1 208
213x3 1 37x2 2 194x 1 208
Chapter 5
136
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5
2
3 2 2i
Theorem Descartes’s 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) 5 0 is either equal to the number of sign
changes between consecutive coefficients of P(x) or less than that by an even number.
The number of negative real roots of P(x) 5 0 is either equal to the number of sign
changes between consecutive coefficients of P(2x) or less than that by an even number.
30. A possible number of positive real roots for x4 2 x3 1 x2 2 x 1 1 is 1 / 2 / 3 / 5 .
Problem 5 Using Descartes’s Rule of Signs
Got It? What does Descartes’s Rule of Signs tell you about the real roots of
2x4 2 x3 1 3x2 2 1 5 0?
31. The number of sign changes is 3, and the number of positive real roots is
4
32. P(2x) 5 2 Q
R 2
3
.
2
1 3Q
The number of sign changes is
or
R 21 5
, and the number of negative real roots is
.
Lesson Check • Do you UNDERSTAND?
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Reasoning In the statement below, r and s represent integers. Is the statement
sometimes, always, or never true? Explain.
A root of the equation 3x3 1 rx2 1 sx 1 8 5 0 could be 5.
33. Write the factors of 8.
34. Write the factors of 3.
35. Because there are / are no factors of 5, the statement
is always / sometimes / never true.
Math Success
Check off the vocabulary words that you understand.
Rational Root Theorem
Conjugate Root Theorem
Descartes’s Rule of Signs
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Lesson 5-5