Rational Zero Theorem

Statement and Application

The Rational Zero Theorem provides a systematic way to list all possible rational zeros of a polynomial function with integer coefficients. If and is a rational zero, then:

• p is a factor of the constant term

• q is a factor of the leading coefficient

Thus, the possible rational zeros are given by:

Example: Listing Possible Rational Zeros

For , the possible rational zeros are:

• Factors of 4 (constant term):

• Factors of -1 (leading coefficient):

• Possible rational zeros:

Example:

• Factors of -3:

• Factors of 4:

• Possible rational zeros:

Finding Zeros of a Polynomial Function

Procedure

To find the zeros of a polynomial function:

1. List all possible rational zeros using the Rational Zero Theorem.

2. Use synthetic division to test each candidate zero.

3. If the remainder is zero, the candidate is a zero of the function.

4. Repeat the process with the reduced polynomial until all zeros are found.

Example:

• Possible rational zeros:

• Test using synthetic division:

Synthetic Division Example

Coefficients: 1, 2, -5, -6

Testing :

• Resulting coefficients: 1, 4, 3, 0

• Remainder is 0, so is a zero.

• Reduced polynomial:

• Factoring:

• Zeros:

Solving Polynomial Equations

Example:

• Possible rational zeros:

• Use synthetic division to test candidates.

After synthetic division, zeros found: (multiplicity 2), and the quadratic yields complex zeros using the quadratic formula:

Linear Factorization Theorem

Statement

If is a polynomial of degree and , then:

• are the complex roots of

Finding a Polynomial Function with Given Zeros

Example

Find a 4th degree polynomial with real coefficients and zeros , and .

• Since is a zero, is also a zero.

• Factors:

• Expanded:

• Using , solve for :

• Final polynomial:

Descartes' Rule of Signs

Determining Number of Real Zeros

Count the number of sign changes in for positive real zeros, and in for negative real zeros. The number of real zeros is equal to or less than the number of sign changes by an even integer.

Example:

• Sign changes in : 3 (so 3 or 1 positive real zeros)

• Sign changes in : 4 (so 4, 2, or 0 negative real zeros)

Example:

• No sign changes in : 0 positive real zeros

• Sign changes in : 3 (so 3 or 1 negative real zeros)

Important Facts about Roots of Polynomials

• If is a polynomial of degree , then has exactly complex roots, counted with multiplicity.

• If is a root, its conjugate is also a root.

• Multiplicity: If a root is repeated, it is counted as many times as its multiplicity.

Synthetic Division

Process Overview

Synthetic division is a simplified method for dividing a polynomial by a linear factor . It is used to test possible rational zeros efficiently.

• Write the coefficients of the polynomial.

• Use the candidate zero outside the division chart.

• Perform the addition and multiplication steps as described above.

• If the remainder is zero, the candidate is a zero.

Quadratic Formula

Solving Quadratic Equations

When a quadratic factor cannot be factored by inspection, use the quadratic formula:

This formula yields real or complex roots depending on the discriminant .

Summary Table: Rational Zero Theorem

Summary Table: Descartes' Rule of Signs