1. Apply the Fundamental Theorem of Algebra Lesson 2.7
Honors Algebra 2
Apply the Fundamental Theorem of Algebra
Lesson 2.7

2. Goals Goal Rubric Level 1 – Know the goals.
Use the Fundamental Theorem of Algebra to find the number of solutions or zeros.
Use the Complex Conjugates Theorem and Irrational Conjugates Theorem to find the zeros of a polynomial function.
Use zeros to write a polynomial function.
Use Descartes’ Rule of Signs to determine the number and type of zeros.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Repeated Solution
Irrational Conjugates
Complex Conjugates

4. Repeated Solutions
The multiplicity of a solution refers to the number of times a solution occurs.
e.g. 𝑥 6 + 𝑥 5 −5 𝑥 4 − 𝑥 3 +8 𝑥 2 −4𝑥=0
𝑥 𝑥 𝑥−1 3 =0
– x = 0 leads to a single solution.
– (x + 2)2 leads to a solution of –2 which occurs twice (a double or repeated solution).
– (x – 1)3 leads to a solution of 1 which occurs three times (also a repeated solution).
So, counting the repeated solutions, the 6th degree polynomial has 6 solutions: 0, –2, –2, 1, 1, 1.

5. Fundamental Theorem of Algebra
The previous result is generalized by the Fundamental Theorem of Algebra.
The corollary to the fundamental theorem of algebra also implies that an nth – degree polynomial function f has exactly n zeros.

6. a. How many solutions does the equation x3 + 5x2 + 4x + 20 = 0 have?
EXAMPLE 1
Find the number of solutions or zeros
a. How many solutions does the equation
x3 + 5x2 + 4x + 20 = 0 have?
SOLUTION
Because x3 + 5x2 + 4x + 20 = 0 is a polynomial
equation of degree 3,it has three solutions.
(The solutions are – 5, – 2i, and 2i.)

7. b. How many zeros does the function f (x) = x4 – 8x3 + 18x2 – 27 have?
EXAMPLE 1
Find the number of solutions or zeros
b. How many zeros does the function
f (x) = x4 – 8x3 + 18x2 – 27 have?
SOLUTION
Because f (x) = x4 – 8x3 + 18x2 – 27 is a polynomial
function of degree 4, it has four zeros. (The zeros
are – 1, 3, 3, and 3.)

8. 1. How many solutions does the equation x4 + 5x2 – 36 = 0 have?
Your turn:
for Example 1
1. How many solutions does the equation
x4 + 5x2 – 36 = 0 have?
ANSWER 4

9. 2. How many zeros does the function f (x) = x3 + 7x2 + 8x – 16 have?
Your Turn:
for Example 1
2. How many zeros does the function
f (x) = x3 + 7x2 + 8x – 16 have?
ANSWER 3

10. Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14.
EXAMPLE 2
Find the zeros of a polynomial function
Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14.
SOLUTION
STEP 1
Find the rational zeros of f. Because f is a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero.
STEP 2
Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x2 – 4x + 7. Therefore:
f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)

11. f(x) = (x + 1)2(x – 2) x – (2 + i 3 ) x – (2 – i 3 )
EXAMPLE 2
Find the zeros of a polynomial function
STEP 3
Find the complex zeros of f . Use the quadratic formula to factor the trinomial into linear factors.
f(x) = (x + 1)2(x – 2) x – (2 + i ) x – (2 – i )
The zeros of f are – 1, – 1, 2, 2 + i , and 2 – i 3.
ANSWER

12. Find all zeros of the polynomial function.
Your Turn:
for Example 2
Find all zeros of the polynomial function.
f (x) = x3 + 7x2 + 15x + 9
The zeros of f are – 1, −3, and – 3.
ANSWER
4. f (x) = x5 – 2x4 + 8x2 – 13x + 6
ANSWER
Zeros of f are 1, 1, – 2, 1 + i 2, and 1 – i 2

13. Behavior Near Zeros
The graph of 𝑓 𝑥 = 𝑥 5 −4 𝑥 4 +4 𝑥 𝑥 2 −13𝑥−14 is shown. The zeros of f are; −1,−1, 2, 2+𝑖 3 ,2−𝑖 3 .
Note that only the real zeros appear as x-intercepts on the graph.
Also note that the graph is tangent to the x-axis at the repeated zero x = −1, but crosses the x-axis at the zero x = 2.
The following figure illustrates some conclusions.

14. Behavior Near Zeros This concept can be generalized as follows:
When a factor x – k of a function f is raised to an odd power, the graph of f crosses the x-axis at x = k.
When a factor x – k of a function f is raised to an even power, the graph of f is tangent to the x-axis at x = k.

15. Complex Conjugates
The function 𝑓 𝑥 = 𝑥 5 −4 𝑥 4 +4 𝑥 𝑥 2 −13𝑥−14 has zeros; −1,−1, 2, 2+𝑖 3 ,2−𝑖 3 .
Notice that the zeros 2+𝑖 3 𝑎𝑛𝑑 2−𝑖 3 are complex conjugates.
A similar result applies to irrational zeros of polynomials functions.
This is generalized in the following theorem.

16. Complex Conjugates

17. EXAMPLE 3
Use zeros to write a polynomial function
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 3 and as zeros.
SOLUTION
Because the coefficients are rational and is a zero, 2 – must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f (x) as a product of three factors.

18. f (x) = (x – 3) [ x – (2 + √ 5 ) ] [ x – (2 – √ 5 ) ]
EXAMPLE 3
Use zeros to write a polynomial function
f (x) = (x – 3) [ x – (2 + √ 5 ) ] [ x – (2 – √ 5 ) ]
Write f (x) in factored form.
= (x – 3) [ (x – 2) – √ 5 ] [ (x – 2) +√ 5 ]
Regroup terms.
= (x – 3)[(x – 2)2 – 5]
Multiply.
= (x – 3)[(x2 – 4x + 4) – 5]
Expand binomial.
= (x – 3)(x2 – 4x – 1)
Simplify.
= x3 – 4x2 – x – 3x2 + 12x + 3
Multiply.
Combine like
terms.
= x3 – 7x2 + 11x + 3

19. f(2 + √ 5 ) = (2 + √ 5 )3 – 7(2 + √ 5 )2 + 11( 2 + √ 5 ) + 3
EXAMPLE 3
Use zeros to write a polynomial function
CHECK
You can check this result by evaluating f (x) at each of
its three zeros.
f(3) = 33 – 7(3)2 + 11(3) + 3 = 27 – = 0 
f(2 + √ 5 ) = (2 + √ 5 )3 – 7(2 + √ 5 )2 + 11( 2 + √ 5 ) + 3
= √ 5 – 63 – 28 √ √ 5 + 3
= 0 
Since f (2 + √ 5 ) = 0, by the irrational conjugates theorem f (2 – √ 5 ) = 0. 

20. Your Turn:
for Example 3
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.
5. – 1, 2, 4
f(x) = x3 – 5x2 + 2x + 8
ANSWER
, 1 + √ 5
f(x) = x3 – 6x2 + 4x +16
ANSWER

21. Your Turn:
for Example 3
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.
, 2i, 4 –
√ 6
f(x) = x5 – 10x4 + 30x3 – 60x x – 80
ANSWER
, 3 – i
f(x) = x3 – 9x2 + 28x –30
ANSWER

22. Descartes’ Rule of Signs
Uses the relationship between the coefficients of a polynomial function and the number of positive and negative zeros of the function.

23. Use Descartes’ rule of signs
EXAMPLE 4
Use Descartes’ rule of signs
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for
f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.
SOLUTION
f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.
The coefficients in f (x) have 3 sign changes, so f has 3 or 1 positive real zero(s).
f (– x) = (– x)6 – 2(– x)5 + 3(– x)4 – 10(– x)3 – 6(– x)2 – 8(– x) – 8
= x6 + 2x5 + 3x4 + 10x3 – 6x2 + 8x – 8

24. Use Descartes’ rule of signs
EXAMPLE 4
Use Descartes’ rule of signs
The coefficients in f (– x) have 3 sign changes, so f has 3 or 1 negative real zero(s) .
The possible numbers of zeros for f are summarized in the table below.

25. Your Turn:
for Example 4
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.
9. f (x) = x3 + 2x – 11
ANSWER

26. Your Turn:
for Example 4
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.
g(x) = 2x4 – 8x3 + 6x2 – 3x + 1
ANSWER

27. Approximating Zeros
Irrational zeros can be approximated using the graphing calculator.
Use the zero feature of the graphing calculator.

28. Approximate the real zeros of
EXAMPLE 5
Approximate real zeros
Approximate the real zeros of
f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.
SOLUTION
Use the zero (or root) feature of a graphing calculator, as shown below.
From these screens, you can see that the zeros are x ≈ – 0.73 and x ≈ 2.73.
ANSWER

29. Approximate the real zeros of f (x) = 3x5 + 2x4 – 8x3 + 4x2 – x – 1.
Your Turn:
for Example 5
Approximate the real zeros of
f (x) = 3x5 + 2x4 – 8x3 + 4x2 – x – 1.
The zeros are x ≈ – 2.2, x ≈ – 0.3, and x ≈ 1.1.
ANSWER

30. EXAMPLE 6
Approximate real zeros of a polynomial model
s (x) = x3 – 0.225x x – 11.0
What is the tachometer reading when the boat travels 15 miles per hour?
A tachometer measures the speed (in revolutions per minute, or RPMs) at which an engine shaft rotates. For a certain boat, the speed x of the engine shaft (in 100s of RPMs) and the speed s of the boat (in miles per hour) are modeled by
TACHOMETER

31. From the graph, there is one real zero: x ≈ 19.9.
EXAMPLE 6
Approximate real zeros of a polynomial model
SOLUTION
Substitute 15 for s(x) in the given function. You can rewrite the resulting equation as:
0 = x3 – 0.225x x – 26.0
Then, use a graphing calculator to approximate the real zeros of f (x) = x3 – 0.225x x – 26.0.
From the graph, there is one real zero: x ≈ 19.9.
The tachometer reading is about 1990 RPMs.
ANSWER

32. 12. WHAT IF? In Example 6, what is the tachometer
Your Turn:
for Example 6
12. WHAT IF? In Example 6, what is the tachometer
reading when the boat travels 20 miles per hour?
The tachometer reading is about 2310 RPMs.
ANSWER