Section 3.3Section 3.3

Theorems about Zeros Theorems about Zeros of Polynomial Functionsof Polynomial Functions

Zeros of a PolynomialZeros of a Polynomial

Zeros (Solutions)

Real Zeros Complex Zeros

Rational ZerosComplex Number and its Conjugate

Fundamental Theorem of AlgebraFundamental Theorem of Algebra

Every polynomial function of degree n with Every polynomial function of degree n with n 1, has at n 1, has at least one complex zeroleast one complex zero..

Complex ZerosComplex Zeros

*** Complex zeros come in pairs. *** Complex zeros come in pairs. ******

Complex conjugates Complex conjugates

a + bi, a - bia + bi, a - bi

Irrational ZerosIrrational Zeros

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

a c b , a c b

Rational Zero TheoremRational Zero Theorem

If the polynomial If the polynomial

P(x) = aP(x) = annxxnn + a + an-1n-1xxn-1n-1 + . . . + a + . . . + a11x + ax + a00

has integer coefficients, then every has integer coefficients, then every rational rational zerozero of P is of the form of P is of the form

where where

p p is a factor of the constantis a factor of the constant coefficient a coefficient a00

and and q q is a factor of the leadingis a factor of the leading coefficient a coefficient ann..

pq

Finding the Rational Zeros of a PolynomialFinding the Rational Zeros of a Polynomial

1.1. List all possible rational zeros of the List all possible rational zeros of the polynomial using the Rational Zero Theorem.polynomial using the Rational Zero Theorem.

2.2. Use synthetic division on each possible Use synthetic division on each possible rational zero and the polynomial until one rational zero and the polynomial until one gives a remainder of zero. gives a remainder of zero. This means you This means you have found a zero, as well as a factor.have found a zero, as well as a factor.

3.3. Write the polynomial as the product of this Write the polynomial as the product of this factor and the quotient.factor and the quotient.

4.4. Repeat procedure on the quotient until the Repeat procedure on the quotient until the quotient is quadratic.quotient is quadratic.

5.5. Once the quotient is Once the quotient is quadraticquadratic, factor or use , factor or use the quadratic formula to find the remaining the quadratic formula to find the remaining real and imaginary zeros.real and imaginary zeros.