1. 7.5.1 Zeros of Polynomial Functions
Objectives:
Use the Rational Root Theorem to find the zeros of a polynomial function

2. Rational Root Theorem
Let f(x) be a polynomial function with integer coefficients in standard form. If (in lowest terms) is a root of f(x) = 0, then
p is a factor of the constant term of f(x)
q is a factor of the leading coefficient of f(x)

3. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 1: Make an organized list of all possible roots.
factors of 2:
factors of 8:
List:
±1, ±½, ±¼, ± ⅛ , ±2

4. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 2: Use substitution or synthetic division to test possible roots, until you find one that works.
List: ±1, ±½, ±¼, ± ⅛ , ±2

5. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 3: Use factoring or quadratic formula to find the other two.
Found: ½ Resulting Equation: 8x2 + 14x – 4 = 0

6. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 2: Use a graphing calculator to identify possible roots.
possible roots: -2

7. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 3: Use substitution or synthetic division to test all possible roots.
possible roots: -2 -2
-16 12 -2 8 -6 1
roots: -2

8. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 3: Use substitution or synthetic division to test all possible roots.
possible roots: -2 2 3 -2 8 12 -8
roots -2

9. Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0.
Step 3: Use substitution or synthetic division to test all possible roots.
possible roots: -2 4 7 -2 8 14 -4
roots: -2

10. Example 2 Find all of the zeros of Q(x) = x3 + 4x2 – 6x - 12.
First, use the Rational Root Theorem and a graph of the polynomial function to determine some possibilities.
Then use synthetic division to test your choices. 2 2 12 12 1 6 6
Since the remainder is 0, x – 2 is a factor of x3 + 4x2 – 6x - 12.

11. Example 2 Find all of the zeros of Q(x) = x3 + 4x2 – 6x - 12.or
Since the remainder is 0, x – 2 is a factor of x3 + 4x2 – 6x - 12.

12. Homework
p.463 #11-21 Odd

13. 7.5.2 Zeros of Polynomial Functions
Objectives:
Use the Complex Conjugate Root Theorem to find the zeros of a polynomial function
Use the Fundamental Theorem to write a polynomial function given sufficient information about its zeros

14. Example 1 Find all of the zeros of P(x) = -4x3 + 2x2 – x + 3.
First, use the Rational Root Theorem and a graph of the polynomial function to determine some possibilities.
Then use synthetic division to test your choices. 1 -4 -2 -3 -4 -2 -3
Since the remainder is 0, x – 1 is a factor of -4x3 + 2x2 – x + 3.

15. Example 1 Find all of the zeros of P(x) = -4x3 + 2x2 – x + 3.or
Since the remainder is 0, x – 1 is a factor of -4x3 + 2x2 – x + 3.

16. Complex Conjugate Root Theorem
If P is a polynomial function with real-number coefficients and a + bi (where b = 0) is a root of P(x) = 0, then a – bi is also a root of P(x) = 0.
Graph R(x) = 2x3 – x2 – 4x. How many real zeros does R have?
Graph S(x) = 2x3 – x2 – 4x How many real zeros does S have?
Graph T(x) = 2x3 – x2 – 4x How many real zeros does T have?

17. Homework
Page 464 #23-33 Odd

18. Example 2
Write a polynomial function, P, given that P has a degree of 3, P(0) = 120, and its zeros are -3, 2, and 4.
Since the zeros are -3, 2, and 4,
x = -3
x = 2
and
x = 4
x + 3= 0
x - 2 = 0
and
x – 4= 0
P(x) = a(x + 3)(x – 2)(x – 4)
P(x) = a(x2 + x - 6)(x – 4)
P(x) = a(x3 - 3x2 – 10x + 24)
120 = a((0)3 - 3(0)2 – 10(0) + 24)
120 = 24 a
a = 5
P(x) = 5(x3 - 3x2 – 10x + 24)
P(x) = 5x3 - 15x2 – 50 x + 120

19. Example 3
Write a polynomial function, P, given that P has a degree of 2, P(0) = 18, and its zeros are i and -3 – 3i.
Since the zeros are i and -3 – 3i,
x = i
and
x = i
x – (-3 + 3i) = 0
and
x – (-3 - 3i) = 0
x i = 0
and
x i = 0
( )( ) = 0
x i
x i
x2 + 3x + 3xi
+ 3x i
– 3xi – 9i – 9i2 = 0
x2 + 6x + 9 – 9(-1) = 0
x2 + 6x + 18 = 0

20. Homework
Page 464 #41-49 Odd