7.5 Roots and Zeros

Objectives: The student will be able to…

1) Determine the number and type of roots for a polynomial equation.

2) Find the zeros of a polynomial function.

Types of Roots• A zero (aka root or solution) of a function

f(x) is any value c such that f(c)=0.

• This means that (x-c) is a factor of f(x).

• the real zeros of the function are the x-intercepts of the graph.

Fundamental Theorem of Algebra

Every polynomial equation with degree greater than zero has at least one root in

the set of complex numbers.

- Imaginary numbers and real numbers both belong to the set of complex numbers.

- Solving polynomial equations may have one or more real roots, or no real roots (roots are imaginary numbers)

Example 1: Solve the equation. State the number and type of roots.

Example 2: Solve the equation. State the number and type of roots.

Example 3: Solve the equation. State the number and type of roots.

Example 4: Solve the equation. State the number and type of roots.

CorollaryA polynomial equation of the form P(x) = 0 of degree n with complex coefficients has exactly n roots in the set of complex numbers.

Descartes’ Rule of SignsIf P(x) is a polynomial with real coefficients whose terms are arranged in descending powers of the variable…

-the number of positive real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms, or is less than this by an even number

AND

-the number of negative real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms of P(-x), or less than this by an even number.

Example 5: State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

Example 6:

Complex Conjugates Theorem

For any polynomial function, if an imaginary number is a zero of that function, its conjugate is also a zero.

Example 7:Write a polynomial function of least degree with integral coefficients that has -4, 1, and 5 as zeros.

Example 8: Write a polynomial function of least degree with integral coefficients whose zeros include 3 and 2-i