2. Objective: To learn & apply the fundamental theorem of algebra & determine the roots of polynomail equations

3. Polynomial – an algebraic expression with more than one term. (Sum of monomials) Each term must be in the form ax n a must be a complex number and n must be a whole number. Ex:2x 3 + 3x +5, 4x 2 – 3x

4. Real Numbers (b is 0) All numbers listed below Rational Numbers Any numbers that can be written as a fraction - 0.3, 2.3, 5, 1/4 Integers Whole numbers and their opposites …-2,- 1,0,1,2,… Whole Numbers 0,1,2,3,… Natural Numbers 1,2,3… Irrational Numbers Nonterminating, nonrepeating decimals , √2 Complex Numbers (a+ bi) Imaginary Numbers (a =0)

5. Leading Coefficient is the “a” on the term with the highest degree. Degree of a Polynomial – is the degree of the term with the highest degree. A function that has a polynomial in one variable is a polynomial function. Any value of x for which f(x) = 0 are called the zeros of the function. When graphing these are also called the x-intercepts. (zeros of functions, roots are for equations)

6. Given the polynomial: P = 3x 3 – 4x 2 + 5x -7 Terms: 3x 3, -4x 2, 5x, -7 Coefficients:3, -4, 5, -7 Degree of each term: 3, 2, 1, 0 Degree of the polynomial: 3 Leading Coefficient: 3

7. Roots and zeros can be real numbers, imaginary numbers or complex numbers.

8. We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n  1, then the equation f (x)  0 has at least one complex root. The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n  1, then the equation f (x)  0 has at least one complex root. The Fundamental Theorem of Algebra

9. The Linear Factor Theorem If f (x)  a n x n  a n  1 x n  1  …  a 1 x  a 0 b, where n  1 and a n  0, then f (x)  a n (x  c 1 ) (x  c 2 ) … (x  c n ) where c 1, c 2,…, c n are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors. The Linear Factor Theorem If f (x)  a n x n  a n  1 x n  1  …  a 1 x  a 0 b, where n  1 and a n  0, then f (x)  a n (x  c 1 ) (x  c 2 ) … (x  c n ) where c 1, c 2,…, c n are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors. Just as an nth-degree polynomial equation has n roots, an nth-degree polynomial has n linear factors. This is formally stated as the Linear Factor Theorem.

10. Given the polynomial function State the degree and the leading coefficient of the polynomial. Determine whether -2 is a zero of f(x).

11. EXAMPLE:Finding a Polynomial Function with Given Zeros Find a fourth-degree polynomial function f (x) with real coefficients that has  2, and i as zeros and such that f (3)   150. Solution Because i is a zero and the polynomial has real coefficients, the conjugate must also be a zero. We can now use the Linear Factorization Theorem.  a n (x  2)(x  2)(x  i)(x  i) Use the given zeros: c 1  2, c 2  2, c 3  i, and, from above, c 4  i. f (x)  a n (x  c 1 )(x  c 2 )(x  c 3 )(x  c 4 ) This is the linear factorization for a fourth-degree polynomial.  a n (x 2  4)(x 2  1) Multiply f (x)  a n (x 4  3x 2  4) Complete the multiplication more

12. Multiplicity refers to the number of times that root shows up as a factor Ex: if -2 is a root with a multiplicity of 2 then it means that there are 2 factors :(x+2)(x+2)

13. Find the polynomial with roots 2, 3i, and -3i Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis?

14. State the number of complex roots of the equation Then find the roots and graph the related function.