2nd Hour Honors Pre-Calculus Winter 2013: Chapter 2.3 (Real Zeros of Polynomial Functions)

Long Division of Polynomials

When using long division to divide polynomials you set it up like any other long division problem. You start with the dividend (numerator) in the box and the divisor (denominator) to the left of the box. Make sure you have all the terms listed, including terms that have a coefficient of 0.

divided by

The quotient is

with a remainder of 1

Synthetic Division

When you are dividing a polynomial by (x-k), where k is any real number, you can use synthetic division.

You set up synthetic division with k to the left of the box and the coefficient of each term in the dividend (including 0s) listed inside the box. Move down the first coefficient and multiply it by k. Then, add this product to the next coefficient in the list. You continue this this until you have used every coefficient in the dividend. The numbers below the line (other than the last one) are the coefficients of each term in the quotient and the number in the bottom right corner is the remainder.

divided by

The quotient is

with a remainder of 1

Theorems

In these theorems, f(x) is a polynomial function

Remainder Theorem

The remainder of f(x) divided by (x-k) is f(k)

Factor Theorem

(x-k) is a factor of f(x) if and only if the remainder is 0

f(k)=0

Rational Root Theorem

Because of the two theorems listed above, we can use synthetic division to factor and find the zeros of polynomial functions. Instead of blindly choosing numbers and checking to see if it is a zero of the function, we can use the rational root theorem to help narrow down possible factors.

k is all possible zeros of the function

FCT is all factors of the constant term

FLC is all factors of the leading coefficient

We can factor the function below and find all real zeros.

-18 is the constant term so the factors include -18, -9, -6, -3, -2, -1, 1, 2, 3, 6, 9, and 18.

2 is the leading coefficient so the factors include -2, -1, 1, and 2.

If we divide any factor of the constant term by any factor of the leading coefficient we will get a possible value of k. All possible values of k include -18, -9, -6, -4.5, -3, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 3, 4.5, 6, 9, and 18.

Using these values we can completely factor the function.