2.1 Zeros of Polynomial Functions

2.1 Zeros of Polynomial Functions

The Fundamental Theorem of Algebra

- If is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system

Study Tip: In order to find the zeros of a function , set it equal to 0 and solve the resulting equation

Example: Find the zeros of the polynomial

- Set the equation equal to zero

- Factor

- Solving each for zero

- Complete factorization

The Rational Zero Test

If the polynomial has integer coefficients, every rational zero of f has the form

Rational zero =

where p and q have no common factors other than 1, and

p = a factor of the constant Factors of the constant term

q = a factor of the leading coefficient Factors of the leading coefficient

EXAMPLE: Find all the rational zeros of the function

- Factors of the constant term are

- Factors of the leading coefficient are

- We can try 1 and use synthetic division and determine that it is a zero

- This leaves us the polynomial

- Factoring this leaves us with

- The rational zeros are therefore Notice these are all possible

EXAMPLE: Factor completely.

- Factors are and

- Possible rational zeros are

- Using the graph we can try -3 and/or 3

- Using synthetic division with 3 we obtain

- Using synthetic division with -3 on the depressed equation we get

- Factoring the last part we get

- This gives is zeros of Notice that these are all possible

Complex Zeros Occur in Conjugate Pairs

Let be a polynomial function that has real coefficients. If , where , is a zero of the function, the conjugate is also a zero of the function.

EXAMPLE: Find a fourth-degree polynomial function with real coefficients that has -1, -1, and 3i as zeros.

- Conjugate pairs says that -3i is a zero

- Foiling gives us

- Finally we multiply and get

EXAMPLE: Find all the zeros of given that 3i is a zero.

- Since 3i is a zero, we also have to have -3i as a zeroConjugate Pairs Theorem

- Foiling yields

- Use long division and divide

- This results in which tells us we have zeros at

EXAMPLE: Find all the zeros of and write the polynomial as a product of linear factors.

- List all the possible rational zeros and giving us

- Using synthetic division we test -2 and get

- Using grouping on the last one yields

- Since we want linear factors, have to factor the last one and we get

giving us zeros at

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