1. Objectives Fundamental Theorem of Algebra 6-6
Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots.
Identify all of the roots of a polynomial equation.
Holt Algebra 2

2. You have learned several important properties about real roots of polynomial equations.
You can use this information to write polynomial function when given in zeros.

3. Example 1: Writing Polynomial Functions2 3
Write the simplest polynomial with roots –1, , and 4.
If r is a zero of P(x), then x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
P(x) = x3 – x2 – 2x + 8 3 11

4. Check It Out! Example 1a
Write the simplest polynomial function with the given zeros.
–2, 2, 4
If r is a zero of P(x), then x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
P(x) = x3– 4x2– 4x + 16

5. Write the simplest polynomial function with the given zeros.
Check It Out! Example 1b
Write the simplest polynomial function with the given zeros.
0, , 3 2 3
If r is a zero of P(x), then x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
P(x) = x3– x2+ 2x 11 3

6. Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

7. Example 2: Finding All Roots of a Polynomial
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
The polynomial is of degree 4, so there are exactly four roots for the equation.
p = –36, and q = 1.
Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.
Find the real roots at or near –1 and 4.

8. Example 2 Continued
Test the possible real roots.

9. Example 2 Continued
The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0.
The solutions are 4, –1, 3i, –3i.

10. Check It Out! Example 2
Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.
Find the real roots at or near –5 and 1.

11. Check It Out! Example 2 Continued
Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots.
Find the real roots at or near –5 and 1.

12. Check It Out! Example 2 Continued
The polynomial factors into (x + 5)(x – 1)(x2 + 4) = 0.
The solutions are –5, 1, –2i, +2i).

13. HOMEWORK- Page 449 # 1-6, 11-19