Recall from our work with rational numbers, that when we divide a sum by a real number, we can use the fact that:\dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}for any real numbers, a, b or c.

We can extend this concept to the division of **polynomial expressions**. For any polynomial expressions A, B or C:\dfrac{A+B}{C} = \dfrac{A}{C} + \dfrac{B}{C}

The simplest form of division of polynomials is when the divisor is a **monomial**. The process involves dividing each term of the polynomial by the monomial, then simplifying each individual fraction using the properties of exponents.

When dividing by a monomial that contains a variable, we use the **quotient rule** of exponents, to simplify each term.

Dividing polynomials involves a process known as algebraic manipulation. We can view the **dividend** as the numerator of a fraction and the **divisor** as the denominator.

By factoring, we can simplify and divide out any factors that are common between the numerator and denominator.

Completely factor both the numerator and denominator.

Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors.

Divide out all common factors from the numerator and denominator.

Simplify the resulting expression.

The rectangle has an area of 4 x^{4} - 12 x square units, and its width is 4x units. Find the length of the rectangle.

Worked Solution

Find the quotient of \left(15s^{4}t^{5}-5s^{3}t^{3}+20s^{2}t\right)\div 40st^{2}.

Worked Solution

Factor the numerators and denominators, then divide the polynomial expressions.

a

\dfrac{2x^{2} + 7x}{6x + 21}

Worked Solution

b

\dfrac{y^{2} + 5y - 24}{y^{3} - 27}

Worked Solution

c

\dfrac{3c^{2}d-9cd}{6cd^{2}+3cd}

Worked Solution

The rectangle shown has an area of 15 n^{3} + 13 n^{2} + 33 n.

a

Find a polynomial expression for its height.

Worked Solution

b

Write two polynomials that could represent finding the area of a triangle with the same dimensions.

Worked Solution

Idea summary

For dividing polynomials where the divisor is a monomial, we can use the fact that \dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}

To divide polynomials with non-monomial divisors:

Completely factor the numerator and denominator

Divide out all common factors between the numerator and denominator

Simplify the resulting expression (if necessary)