# Dividing Polynomials

In this section, you’ll learn two methods for dividing
polynomials , long division and

synthetic division . You’ll also learn two theorems that will allow you to
interpret results

when you divide.

Suppose P(x) and D(x) are polynomial functions and D(x) ≠ 0. Then there
are unique

polynomials Q (x) (called the quotient) and R(x) (called the remainder) such that

P(x) = D(x) *Q(x) + R(x) . We call D(x) the divisor.

The remainder function, R(x) , is either 0 or of degree less than the degree of
the divisor.

You can find the quotient and remainder using long division. Recall the steps
you

learned in elementary school to perform long division:

**Example 1:** Find

You can use the same steps when using long division to
divide polynomials.

**Example 2:** Find the quotient and the remainder using long division:

**Example 3: **Use long division to find the quotient
and the remainder:

**Example 4:** If D(x) = 2x - 5 , Q(x) = 3x^{2} + 5x and
R(x) =12 , find P(x).

Often it will be more convenient to use synthetic division to divide
polynomials. This

method is easy to use, as long as your divisor is x ± c , for any real number c.

Synthetic division is best demonstrated by example, so here are a few:

**Example 5:** Find the quotient and the remainder using synthetic division:

**Example 6:** Find the quotient and the remainder using
synthetic division:

**Example 7:** Find the quotient and the remainder using
synthetic division:

**Example 8: **Find the quotient and the remainder using
synthetic division:

Here are two theorems that can be helpful when working
with polynomials:

**The Remainder Theorem:** If P(x) is divided by x - c , then the remainder is P(c)
.

**The Factor Theorem :** c is a zero of P(x) if and only if x - c is a factor of P(x)
, that is

if the remainder when dividing by x - c is zero .

You can use synthetic division and the remainder theorem to evaluate a function
at a

given value .

**Example 9: **Use synthetic division and the remainder theorem to find P(3) for

**Example 10: **Use synthetic division and the remainder
theorem to find P(-1) for

**Example 11: **Determine if x + 2 is a factor of

**Example 12:** Show that x = -1is a zero of
and find the

remaining zeros of the function .

**Example 13: **Show that x = 2 and x = -3are zeros of

and find the remaining zeros of the function .

Finally, you’ll need to work backwards to write a
polynomial with given zeros.

**Example 14:** Find a polynomial of degree 4 with zeros at -3, -1, 2 and 5.

**Example 15: **Find a polynomial of degree 3 with zeros at 0,
2 and -3.

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