When we first begin our Mathematical Journey in Primary School, we cover the basics of division - specifically long division of large number. (If you forgot i recommend re-visiting the theory as this topic is very similar).

We know that in many cases, a quadratic expression can be factorised into two linear factors (Distinct or Equal). Sometimes an exam question may require you to divide a polynomial by another polynomial. For the level expected in this note - you will only be asked to divide a polynomial by a LINEAR expression, for example:

Divide \(x^2 - 2x +1\) by \( x+1 \).

This may be written as:

Let's dive into Long Division of a Polynomial and a Linear Expression.

How to do it!

• Firstly, place your expressions in the following form (should be familiar from primary school):

• Now, we focus on the first 'x' variables in each expression. We divide the numerator's first term by the denominator's first term. \( \cfrac{x^2}{x} \)

• Next, we multiply the quotient of that operation by each value of the denominator.

• Subtract the expressions. Bring down the remaining parts of the expression.

• We have a new expression at the bottom of our working, we repeat the process using this new expression as our numerator and the same linear expression as our denominator. Focus on the first variables - divide them.

• Multiply the new quotient value by all terms in the denominator value.

• Subtract the expressions.

• We keep repeating these steps until we can no longer divide the numerator by the denominator. Sometimes our operations will result in zero (meaning the quotient is a factor of the numerator expression) or there may remain a numerical value (meaning there is a remainder to the division process).

So this is our entire operation:

Dealing with a numerator / denominator with 'zeroed' terms

• When you are given en expression with missing or 'zeroes' terms - your must include these missing variables with a coefficient of zero (0).

For instance, \(x^3 + x - 1 \) - the \(x^2 \) term is missing.

Hence, we will use the expression \( x^3 + 0x^2 + x - 1 \) in our operations.

Writing your solution after the tiring long division process

Let us re-visit primary school mathematics, take a look at the long divison with a remainder below.

Notice how we write the solution?

Same applies for Polynomial Long Division.

Now you know how to long divide in the world of polynomials!

If you have any questions feel free to reach out.