Section 3.3 Theorems about Zeros of Polynomial Functions.

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Presentation transcript:

Section 3.3 Theorems about Zeros of Polynomial Functions

Zeros of a Polynomial Zeros (Solutions) Real Zeros Rational Zeros Complex Zeros Complex Number and its Conjugate

Fundamental Theorem of Algebra  Every polynomial function of degree n with n 1, has at least one complex zero.

Complex Zeros *** Complex zeros come in pairs. *** Complex conjugates a + bi, a - bi

Irrational Zeros *** Irrational zeros come in pairs. ***

Rational Zero Theorem If the polynomial If the polynomial P(x) = a n x n + a n-1 x n a 1 x + a 0 P(x) = a n x n + a n-1 x n a 1 x + a 0 has integer coefficients, then every rational zero of P is of the form has integer coefficients, then every rational zero of P is of the form where where p is a factor of the constant coefficient a 0 p is a factor of the constant coefficient a 0 and q is a factor of the leading coefficient a n. and q is a factor of the leading coefficient a n.

Finding the Rational Zeros of a Polynomial 1. List all possible rational zeros of the polynomial using the Rational Zero Theorem. 2. Use synthetic division on each possible rational zero and the polynomial until one gives a remainder of zero. This means you have found a zero, as well as a factor. 3. Write the polynomial as the product of this factor and the quotient. 4. Repeat procedure on the quotient until the quotient is quadratic. 5. Once the quotient is quadratic, factor or use the quadratic formula to find the remaining real and imaginary zeros.