Find a polynomial with specified zeros. For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible. Use Descartes’ rule of signs to find information about the number of real zeros of a polynomial function with real coefficients.

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4.4 Theorems about Zeros of Polynomial Functions

The Fundamental Theorem of Algebra

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Every polynomial function of degree n, with n 1, has at least one zero in the system of complex numbers.

The Fundamental Theorem of Algebra

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Example: Find a polynomial function of degree 4 having zeros 1, 2, 4i, and 4i.

( ) ( 1)( 2)( 4 )( 4 )nf x a x x x i x i

Zeros of Polynomial Functions with Real Coefficients

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Nonreal Zeros: If a complex number a + bi, b 0, is a zero of a polynomial function f(x) with real coefficients, then its conjugate, a bi, is also a zero. (Nonreal zeros occur in conjugate pairs.)

Irrational Zeros: If where a, b, and c are rational and b is not a perfect square, is a zero of a polynomial function f(x) with rational coefficients, then its conjugate is also a zero.

, a c b

,a c b

Example

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Suppose that a polynomial function of degree 6 with rational coefficients has 3 + 2i, 6i, and as three of its zeros. Find the other zeros.

1 2

Rational Zeros Theorem

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1 21 2 1 0( ) ... ,n n

n nP x a x a x a x a x a Let

where all the coefficients are integers. Consider a rational number denoted by p/q, where p and q are relatively prime (having no common factor besides 1 and 1). If p/q is a zero of P(x), then p is a factor of a0 and q is a factor of an.

Example

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Given f(x) = 2x3 3x2 11x + 6:

a) Find the rational zeros and then the other zeros.

b) Factor f(x) into linear factors.

3 31 12 2 2 2

1, 2, 3, 6:

1, 2

/ : 1, 1,2, 2,3, 3,6, 6, , , ,

Possibilities for p

Possibilities for q

Possibilities for p q

More Examples

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