Fundamental Theorem of Algebra

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Presentation transcript:

Fundamental Theorem of Algebra Unit 2 -- Polynomials Fundamental Theorem of Algebra

Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y. The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

Factors, Roots, Zeros For our Polynomial Function: The Factors are: (x + 5) & (x - 3) The Roots/Solutions are: x = -5 and 3 The Zeros are at: (-5, 0) and (3, 0)

Roots & Zeros of Polynomials II Finding the Roots/Zeros of Polynomials: The Fundamental Theorem of Algebra Descartes’ Rule of Signs The Complex Conjugate Theorem

Fundamental Theorem Of Algebra Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers. A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers. COROLLARY:

Real/Imaginary Roots If a polynomial has ‘n’ complex roots will its graph have ‘n’ x-intercepts? By Fund Theorem of Alg Corollary, degree n = 3 means 3 roots in the complex system. Factor and solve the 3 roots x = 0, x = 2 and x = -2. 3 x-intercepts

Real/Imaginary Roots Just because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real! Again, by Fund Theorem of Alg Corollary, degree n = 3 means 3 roots in the complex system. How many x-intercepts? What are the other roots?

Descartes’ Rule of Signs Arrange the terms of the polynomial P(x) in descending degree: The number of times the coefficients of the terms of P(x) change sign = the number of Positive Real Roots (or less by any even number) The number of times the coefficients of the terms of P(-x) change sign = the number of Negative Real Roots (or less by any even number) In the examples that follow, use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

Find Roots/Zeros of a Polynomial We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring. Some are easier to factor than others! The roots are: 0, -2, 2

Complex Conjugates Theorem If complex roots/zeros are not Real, then they have an Imaginary component. Complex roots with Imaginary components always exist in Conjugate Pairs. If a + bi (b ≠ 0) is a zero of a polynomial function, then its Conjugate, a - bi, is also a zero of the function.

Find Roots/Zeros of a Polynomial If you know one imaginary root, you can use the Complex Conjugates Theorem to determine another root. Ex: Find all the roots of If one root is 4 - i. By Complex Conjugate Theorem, we know 4 + i is another root.

Example (con’t) Ex: Find all the roots of The root 4 – i has the factor [x - (4 - i)]. The root 4 + i has the factor [x - (4 + i)]. Multiply these factors:

Example (con’t) Ex: Find all the roots of If one root is 4 - i. If the product of the two non-real factors is then the third factor (that gives us the neg. real root) is the quotient of P(x) divided by : The third root is x = -3

Complex Zeros; Fundamental Theorem of Algebra Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Find the remaining complex zeros of the given polynomial functions

Complex Zeros; Fundamental Theorem of Algebra Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Long Division

Complex Zeros; Fundamental Theorem of Algebra Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Examples 4) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What is the function?