In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we'll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we'll start with an Example to review fraction addition.

The sum, \(\dfrac{y}{5}+\dfrac{2}{5}\),
simplifies to \(\dfrac{y+2}{5}\).

Table 6.2

Now we will do this in reverse to split a single fraction into separate fractions.

We'll state the fraction addition property here just as you learned it and in reverse.

Fraction Addition

If \(a, b\), and \(c\) are numbers where \(c \neq 0\), then

\(\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}\) and \(\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}\)

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

For example, \(\dfrac{y+2}{5}\)
can be written \(\dfrac{y}{5}+\dfrac{2}{5}\).

Table 6.3

We use this form of fraction addition to divide polynomials by monomials.

Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example 6.77

Find the quotient: \(\dfrac{7 y^{2}+21}{7}\).

Solution
  \(\dfrac{7 y^{2}+21}{7}\)
Divide each term of the numerator by the denominator. \(\dfrac{7 y^{2}}{7}+\dfrac{21}{7}\)
Simplify each fraction. \(y^{2}+3\)
Try It 6.153

Find the quotient: \(\dfrac{8 z^{2}+24}{4}\).

Try It 6.154

Find the quotient: \(\dfrac{18 z^{2}-27}{9}\).

Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

Example 6.78

Find the quotient: \(\left(18 x^{3}-36 x^{2}\right) \div 6 x\).

Solution
  \(\left(18 x^{3}-36 x^{2}\right) \div 6 x\)
Rewrite as a fraction. \(\dfrac{18 x^{3}-36 x^{2}}{6 x}\)
Divide each term of the numerator by the denominator. \(\dfrac{18 x^{3}}{6 x}-\dfrac{36 x^{2}}{6 x}\)
Simplify. \(3 x^{2}-6 x\)
Try It 6.155

Find the quotient: \(\left(27 b^{3}-33 b^{2}\right) \div 3 b\).

Try It 6.156

Find the quotient: \(\left(25 y^{3}-55 y^{2}\right) \div 5 y\).

When we divide by a negative, we must be extra careful with the signs.

Example 6.79

Find the quotient: \(\dfrac{12d^2−16d}{−4}\).

Solution
  \(\dfrac{12 d^{2}-16 d}{-4}\)
Divide each term of the numerator by the denominator. \(\dfrac{12 d^{2}}{-4}-\dfrac{16 d}{-4}\)
Simplify. Remember, subtracting a negative is like adding a positive! \(-3 d^{2}+4 d\)
Try It 6.157

Find the quotient: \(\dfrac{25y^{2}-15y}{-5}\).

Try It 6.158

Find the quotient: \(\dfrac{42b^{2}-18b}{-6}\).

Example 6.80

Find the quotient: \( \dfrac{25y^2−75y^3}{5y^2} \).

Solution
  \(\dfrac{105 y^{5}+75 y^{3}}{5 y^{2}}\)
Separate the terms. \(\dfrac{105 y^{5}+75 y^{3}}{5 y^{2}}\)
Simplify. \(21 y^{3}+15 y\)
Try It 6.159

Find the quotient: \(\dfrac{60d^7−24d^5}{4d^3}\).

Try It 6.160

Find the quotient: \( \dfrac{216p^7-48p^5}{6p^3} \).

Example 6.81

Find the quotient: \((15x^3y−35xy^2)÷(−5xy)\).

Solution
  \(\left(15 x^{3} y-35 x y^{2}\right) \div(-5 x y)\)
Rewrite as a fraction. \(\dfrac{15 x^{3} y-35 x y^{2}}{-5 x y}\)
Separate the terms. \(\dfrac{15 x^{3} y}{-5 x y}-\dfrac{35 x y^{2}}{-5 x y}\)
Simplify. \(-3 x^{2}+7 y\)
Try It 6.161

Find the quotient: \((32a^2b−16ab^2)÷(−8ab)\).

Try It 6.162

Find the quotient: \((−48a^8b^4−36a^6b^5)÷(−6a^3b^3)\).

Example 6.82

Find the quotient: \(\dfrac{36 x^{3} y^{2}+27 x^{2} y^{2}-9 x^{2} y^{3}}{9 x^{2} y}\).

Solution
  \(\dfrac{36 x^{3} y^{2}+27 x^{2} y^{2}-9 x^{2} y^{3}}{9 x^{2} y}\)
Separate the terms. \(\dfrac{36 x^{3} y^{2}}{9 x^{2} y}+\dfrac{27 x^{2} y^{2}}{9 x^{2} y}-\dfrac{9 x^{2} y^{3}}{9 x^{2} y}\)
Simplify. \(4 x y+3 y-y^{2}\)
Try It 6.163

Find the quotient: \(\dfrac{40 x^{3} y^{2}+24 x^{2} y^{2}-16 x^{2} y^{3}}{8 x^{2} y}\).

Try It 6.164

Find the quotient: \(\dfrac{35 a^{4} b^{2}+ 14 a^{4} b^{3}- 42a^{2} b^{4}}{7a^{2} b^2}\).

Example 6.83

Find the quotient: \(\dfrac{10x^2+5x−20}{5x}\).

Solution
  \(\dfrac{10 x^{2}+5 x-20}{5 x}\)
Separate the terms. \(\dfrac{10 x^{2}}{5 x}+\dfrac{5 x}{5 x}-\dfrac{20}{5 x}\)
Simplify. \(2 x+1+\dfrac{4}{x}\)
Try It 6.165

Find the quotient: \(\dfrac{18c^2+6c−9}{6c}\).

Try It 6.166

Find the quotient: \(\dfrac{10d^2−5d−2}{5d}\).

Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/6-6-divide-polynomials
Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License.

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