Roots & Zeros of Polynomials II

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Roots & Zeros of Polynomials II Finding the Roots/Zeros of Polynomials: The Fundamental Theorem of Algebra, Descartes’ Rule of Signs, The Complex Conjugate Theorem Created by K. Chiodo, HCPS

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? In this example, the degree is n = 3, and if we factor the polynomial, the roots are x = -2, 0, 2. We can also see from the graph that there are three x-intercepts.

Real/Imaginary Roots Just because a polynomial has ‘n’ complex roots does not mean that they are all Real! In this example, however, the degree is still n = 3, but there is only one real x-intercept or root at x = -1, the other 2 roots must have imaginary components.

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 signs equals 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 signs equals 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 zero and factoring. Some are easier to factor than others! The roots are: 0, -2, 2 .

Find Roots/Zeros of a Polynomial If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial. The resulting polynomial has a lower degree and might be easier to factor or solve with the quadratic formula. (x - 5) is a factor We can solve the resulting polynomial to get the other 2 roots:

Complex Conjugates Theorem Roots/Zeros that are not Real are Complex with 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 the known root is imaginary, we can use the Complex Conjugates Theorem. Ex: Find all the roots of if one root is 4 - i. Because of the Complex Conjugate Theorem, we know that another root must be 4 + i. Can the third root also be imaginary? Consider… Descartes: # of Pos. Real Roots = 2 or 0 Descartes: # of Neg. Real Roots = 1

Example (con’t) Ex: Find all the roots of If one root is 4 - i. If one root is 4 - i, then one factor is [x - (4 - i)], and another root is 4 + i, with the other factor as [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 is the quotient of P(x) divided by x 2 – 8x + 17. The third root is x = -3

Finding Roots/Zeros of Polynomials We use the Fundamental Theorem of Algebra, Descartes’ Rule of Signs and the Complex Conjugate Theorem to predict the nature of the roots of a polynomial. We use skills such as factoring, polynomial division and the quadratic formula to find the zeros/roots of polynomials. In future lessons you will learn other rules and theorems to predict the values of roots so you can solve higher degree polynomials!