Fundamental Theorem of Algebra and Finding Real Roots

1. Fundamental Theorem of Algebra and Finding Real Roots Honors Advanced Algebra Presentation 2-6

2. Warm-Up • Given the roots, write the factors of the quadratic and the polynomial. • x = -3, 2 • x = 4, 3 • x = -5, 5 • x = 1, 1

3. Roots of a Polynomial • The places where a polynomial crosses or touches the x-axis are called the roots of the polynomial. • They are also known as x-intercepts, zeros, or solutions. • Roots can be found by setting the polynomial equal to 0 and solving.

4. The Fundamental Theorem of Algebra • Every polynomial function of degree has at least one zero, where a zero may be a complex number. • Corollary: Every polynomials function of degree has exactly n zeros, including multiplicities and irrational roots.

5. Multiplicity of a Root • A root can occur once or multiple times. • If the root repeats, the number of times is known as the multiplicity of a root. • If the multiplicity is even, the graph will touch the x-axis but not cross it; if the multiplicity is odd, the graph will intersect the x-axis. • Example: Write a polynomial with roots 1 with a multiplicity of 2 and -3 with a multiplicity of 1.

6. Multiplicity of a Root • Multiplicity of 1 • Multiplicity of 2 • Multiplicity of 3

7. Multiplicity of a Root • Example 2: Write a possible polynomial as the product of factors given the graph below.

8. Rational Roots Theorem • If the polynomial P(x) has integer coefficients, then every rational root of the polynomial equation P(x) = 0 can be written in the form , where p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

9. Rational Roots Theorem • Example: Find all possible rational roots of

10. Rational Roots Theorem • Example: Use the rational roots theorem and a graph to completely factor

11. Irrational Roots Theorem • If the polynomial P(x) has rational coefficients and is a root of the polynomial equation P(x) = 0, where a and b are rational and is irrational, then is also a root of P(x) = 0.

12. Irrational Roots Theorem • Irrational roots always come in conjugate pairs. • Example: If 5 is a root of a polynomial, what is another root of the polynomial? What is the corresponding factor?

13. Complex Conjugate Root Theorem • If a + bi is a root of a polynomial equation with real-number coefficients, then a – bi is also a root. • Imaginary roots always come in conjugate pairs. • Example: If 3 + 2i is a root of a polynomial, what is another root of the polynomial? What is the corresponding factor?

14. Example of Finding Roots • Solve by finding all roots. • Step 1: Use the Rational Roots Theorem to identify possible rational roots.

15. Example of Finding Roots • Solve by finding all roots. • Step 2: Graph the polynomial to narrow down your options.

16. Example of Finding Roots • Solve by finding all roots. • Step 3: Test the possible real roots to help factor the polynomial down to a quadratic. Then find remaining zeros.

17. Example of Writing a Polynomial Function given Zeros • Write the simplest polynomial function with zeros 1 + i, , and -3.

18. Homework • P. 121, #24-25, 30-35 • Pg. 127-128, #12-32 even, 38-43