1. Do Now: Find all real zeros of the function.

2. 5.7: Apply the Fundamental
Academy Algebra II
5.7: Apply the Fundamental
Theorem of Algebra
HW Friday: p.384 (12-18 even)
Test : Tuesday, 12/2

3. Descartes’ Rule of Signs
The number of positive real zeros of f is equal to the number of sign changes in the 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.

4. Fundamental Theorem of Algebra
If f(x) is a polynomial with degree of n (where n>0), then the equation f(x) = 0 has at least one solution.
Corollary: The equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

5. Fundamental Theorem of Algebra
Example: x3 – 5x2 – 8x + 48 = 0
(x+3)(x – 4)2= 0
x = -3 and x = 4
This equation only has two distinct solutions: -3 and 4.
Because the factor (x – 4) appears twice, you can count the solution 4 twice. With the repeated solution the third-degree equation has three solutions.

6. Solve the equation.
x3 + 5x2 + 4x + 20 = 0

7. Find the zeros of the polynomial function.
f(x) = x4 – 8x3 + 18x2 – 27

8. Find all zeros of the polynomial function.
f(x) = x5 – 2x4 + 8x2 – 13x + 6

9. Conjugates Theorem
If is an imaginary zero of function f, then is also a zero.
If is a zero of function f where is irrational, then is also a zero.

10. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, & the given zeros.
1.) -1, 2, 4 2.) 4,

11. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, & the given zeros.
3.) 2, 2i,