1. 2.7 Apply the Fundamental Theorem of Algebra day 2
How do you use zeros to write a polynomial function?
What is Descartes’ rule of signs?

2. 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 the given zeros – 1, 2, 4.
Use the three zeros and the factor theorem to write f(x) as a product of three factors.
SOLUTION
x=−1, x+1=0, x=2, x−2=0, x=4, x−4=0
f (x) = (x + 1) (x – 2) ( x – 4)
Write f (x) in factored form.
= (x + 1) (x2 – 4x – 2x + 8)
Multiply.
= (x + 1) (x2 – 6x + 8)
Combine like terms.
= x3 – 6x2 + 8x + x2 – 6x + 8
Multiply.
Combine like terms.

3. Complex Conjugates theorem
If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a −bi is also a zero of f. Imaginary numbers always travel in pairs!

4. Using Zeros to Write Polynomial Functions
Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 2 and 1 + i as zeros.
x = 2, x = 1 + i, AND x = 1 − i.
Complex conjugates always travel in pairs.
f(x) = (x − 2)[x − (1 + i )][x − (1 − i )]
f(x) = (x − 2)[(x − 1) − i ][(x − 1) + i ]
f(x) = (x − 2)[(x − 1)2 − i2 ]
f(x) = (x − 2)[(x2 − 2x + 1 −(−1)]
f(x) = (x − 2)[x2 − 2x + 2]
f(x) = x3 − 2x2 +2x − 2x2 +4x − 4
f(x) = x3 − 4x2 +6x − 4

5. Irrational Conjugates Theorem

6. Use zeros to write a polynomial function
SOLUTION

7. Write f (x) in factored form.
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

8. Write f (x) in factored form. = f(x) =(x – 3)[x – (3 – i)][x –(3 + i)]
Because the coefficients are rational and 3 –i is a zero, 3 + i must also be a zero by the complex conjugates theorem. Use the three zeros and the factor theorem to write f(x) as a product of three factors
SOLUTION
Write f (x) in factored form.
= f(x) =(x – 3)[x – (3 – i)][x –(3 + i)]
= (x–3)[(x– 3)+i ][(x2 – 3) – i]
Regroup terms.
= (x–3)[(x – 3)2 –i2)]
Multiply.
= (x– 3)[(x – 3)+ i][(x –3) –i]
= (x – 3)[(x – 3)2 – i2]=(x –3)(x2 – 6x + 9)
= (x–3)(x2 – 6x + 9)
Simplify.
= x3–6x2 + 9x – 3x2 +18x – 27
Multiply.
= x3 – 9x2 + 27x –27
Combine like terms.

9. Descartes’ Rule of Signs
French mathematician Rene Descartes ( ) found a relationship between the coefficients of a polynomial functions and the number of positive and negative zeros of the function.

10. Descartes’ Rule of Signs

11. 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

12. 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.

13. Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function.
9. f (x) = x3 + 2x – 11
SOLUTION
f (x) = x3 + 2x – 11
The coefficients in f (x) have 1 sign changes, so f has 1 positive real zero(s).

14. f (– x) = (– x)3 + 2(– x) – 11
= – x3 – 2x – 11
The coefficients in f (– x) have no sign changes.
The possible numbers of zeros for f are summarized in the table below.

15. g(x) = 2x4 – 8x3 + 6x2 – 3x + 1
SOLUTION
f (x) = 2x4 – 8x3 + 6x2 – 3x + 1
The coefficients in f (x) have 4 sign changes, so f has 4 positive real zero(s).
f (– x) = 2(– x)4 – 8(– x)3 + 6(– x)2 + 1
= 2x4 + 8x + 6x2 + 1
The coefficients in f (– x) have no sign changes.
The possible numbers of zeros for f are summarized in the table below.

16. How do you use zeros to write a polynomial function?
If x = #, it becomes a factor (x ± #). Multiply factors together to find the equation.
What is Descartes’ rule of signs?
The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(−x) or is less than this by an even number.

17. Assignment is p. 141, 20-28 even,34-40 even
Show your work
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