Section 4.2 – Dividing Polynomials 1

Section 4.2

Dividing Polynomials

Long Division of Polynomials

The division algorithm states that if 𝑃(𝑥) and 𝐷(𝑥) are polynomials, with 𝐷(𝑥) ≠ 0,

then there are unique polynomials 𝑄(𝑥) and 𝑅(𝑥) such that

𝑃(𝑥) = 𝐷(𝑥) ∙ 𝑄(𝑥) + 𝑅(𝑥).

The polynomial 𝑃(𝑥) is called the dividend, 𝐷(𝑥) is called the divisor, 𝑄(𝑥) is called the

quotient and 𝑅(𝑥) is called the remainder. The remainder 𝑅(𝑥) is either 0, or a

polynomial of degree less than the degree of the divisor 𝐷(𝑥).

Note: If the remainder 𝑅(𝑥) = 0, then we say 𝑃(𝑥) is divisible by 𝐷(𝑥) and 𝑄(𝑥) is a

factor of the polynomial 𝑃(𝑥), i.e. 𝑃(𝑥) = 𝐷(𝑥) ∙ 𝑄(𝑥).

Steps to Dividing Polynomials Using Long Division

We divide a polynomial by another polynomial in much the same way we perform long

division with numbers:

1. Write the terms of the dividend and divisor in descending order of the exponents.

2. If there are any missing powers of x, add them in with a coefficient of 0.

3. Divide.

4. The process stops when degree of the remainder is smaller than degree of the divisor.

Example 1: Divide the following polynomials:

a. 8𝑥2+6𝑥−25

4𝑥+9

Section 4.2 – Dividing Polynomials 2

b. 2𝑥3−10𝑥2−3𝑥−10

2𝑥2−3

c. 5

1532

34

x

xx

d. 4

40183 23

x

xxx

Section 4.2 – Dividing Polynomials 3

Dividing Polynomials Using Synthetic Division

We can use synthetic division to divide polynomials if the divisor is of the form x c.

Steps to Dividing Polynomials Using Synthetic Division

1. Write the terms of the dividend and divisor in descending order of the exponents.

2. If there are any missing powers of x, add them in with a coefficient of 0.

3. List the coefficients of the dividend.

4. For dividing by x – c, place c to the left of the coefficients.

For dividing by x + c, place –c to the left of the coefficients.

5. Bring down the first coefficient.

6. Multiply by c and add, for each column.

7. Read the quotient and remainder from the bottom row.

Note: The degree of the quotient will be one less than the degree of the dividend.

Example 2: Divide the following polynomials using synthetic division:

a. 4

40183 23

x

xxx

b. −𝑥3+6𝑥2+2𝑥−4

𝑥−1

Section 4.2 – Dividing Polynomials 4

c. 3𝑥5−7𝑥3−8

𝑥+2

The Remainder Theorem

If the polynomial P(x) is divided by x – c, then the remainder is P(c).

The Factor Theorem

𝒄 is a zero of the polynomial 𝑃(𝑥) if and only if (𝑥 − 𝑐) is a factor of 𝑃(𝑥).

Example 3: Use synthetic division and the Remainder Theorem to evaluate P(-4):

1001122)( 23 xxxxP

Section 4.2 – Dividing Polynomials 5

Example 4: Use synthetic division and the Remainder Theorem to evaluate P(2):

25894)( 234 xxxxxP

Example 5: Determine if (𝑥 + 2) is a factor of 10x3x6x)x(P23 .

Example 6: Show that 𝑥 = −1 is a zero of 15x3x15x3)x(P23 .

Find the remaining zeros of the function.

Section 4.2 – Dividing Polynomials 6

Example 7: Show that 𝑥 = 2 and 𝑥 = −3 are zeros of 24x26x3x6x)x(P234

Find the remaining zeros of the function.

Example 8: Find the 3rd degree polynomial 𝑝(𝑥) with integer coefficients given that

0, 2 and - 3 are zeros, such 𝑝(1) = 8.