In the previous section, we covered dividing monomials and polynomials by monomials. Now, let’s explore the methods for Dividing a Polynomial by another Polynomial. This requires more advanced factorisation skills or the use of the Long Division method.

(iii) Division of a Polynomial by another Polynomial Using Factorisation

This method works efficiently when the polynomial in the numerator can be factored to include the denominator as one of its factors.

Examples:

1) Divide $(5z^2 – 80)$ by $5(z+4)$

Solution:

First, focus on factorising the numerator, $5z^2 – 80$:

1. Take out the Common Factor:
   $$5z^2 – 80 = 5(z^2 – 16)$$
2. Apply the Difference of Squares Identity ($\mathbf{a^2 – b^2 = (a-b)(a+b)}$):
   $$5(z^2 – 4^2) = 5(z – 4)(z + 4)$$

Now, set up the division and cancel the common factors:

2) Divide $35a^2 + 32a – 99$ by $7a – 9$

Solution:

Here, we must factor the quadratic polynomial in the numerator, $35a^2 + 32a – 99$, by splitting the middle term.

1. Split the Middle Term: We need two numbers that multiply to $(35 \times -99 = -3465)$ and add up to $32$. These numbers are $77$ and $-45$.
   $$35a^2 + 32a – 99 = 35a^2 + 77a – 45a – 99$$
2. Group and Factorise:
   $$\begin{aligned} &= (35a^2 + 77a) – (45a + 99) \ &= 7a(5a + 11) – 9(5a + 11) \ &= (7a – 9)(5a + 11)\end{aligned}$$

Now, set up the division and cancel the common factor: