Section 2.3 Real Zeros of Polynomial Functions

There are two ways to divide a polynomial function.
One of them is long division. Long division is harder than synthetic division.

Here is an example of long division.

The second way to divide a polynomial function is known as synthetic division. Synthetic division is easier than long division.

Here are two examples of synthetic division. The zero of this equation would be -4.

To make synthetic division easier, we use the rational zero test (also known as the rational root theorem). This relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term of the polynomial.

Example 7 on page 166 in the book.

Find the rational zeros of f(x) = x^3+x+1

Because the leading coefficient is 1, the possible rational zeros are simply on the factors of the constant term.

Possible Rational Zeros: ±1

f(1) = (1)^3+1+1 = 3

f(-1) (-1)^3+(-1)+1 = -1

The polynomial has no rational zeros.