Let's revise how to divide by polynomials with tips and examples in A-Level Maths!

**Dividing polynomials by ****(x ± p)**

The question: “Divide x^{3} + 6x^{3} + 8x + 3 by ( x +1) ”

Tthe bestway to tackle this is to explain by example…:

Explaining the colours:

📐 Black and pinkis the original sum bit polynomial by x.

To work out what to divide the pink bit by, you divide the first term of the polynomial by x. (The highlighted yellow thing).

📐 Pale purple is the first step:

o You multiply (x + 1) by x^{2 }to give x^{3} + x^{2 }. You then take this away from x^{3} + 6x^{2} +8x + 3

📐 This gives you 5x^{2 }. You then bring down the 8x to give you 5x^{2} + 8x

o Multiply (x + 1) by 5x to give you 5x^{2}+ 5x. As before, you then take this away from 5x^{2} + 8x…

📐 To give 3x. Bring down the 3, and this gives you 3x + 3.

o Multiply (x + 1) by 3 to give 3x +3. Take it away, and you get 0.

AND YOU’RE DONE!

Your final answer is the bit that’s at the top of the bar thingy, and is called the quotient.

Be warned - they’re gunna get harder.

- Sometimes, you’ll get numbers leftover at the end – i.e. you won’t get zero at the end.
- Also, you’ll probably have to divide by x minus something – in which case, you’ll have to be aware of whether you multiply the polynomial by a positive or a negative number.
- You need to remember this whatever the question – the polynomial should always be in the form:

**ax**^{4}** + bx**^{3}** + cx**^{2}** + dx+ e **

- So that means if you get something like: 2x
^{3}+ 5x + 3 , you should put in the x2 term, like so: 2x^{3}+ 0x^{2}+5x +3 - But above all, the key is to just stay calm – once you know the basics for this type of question, you’re pretty much set!

Below are a few questions for you to try and on the next page a few worked answers. Enjoy!

1. Divide x^{3} + 10x^{2}+ 25x + 4 by (x + 4)

2. Divide x^{3} –x^{2} + x + 14 by (x + 2)

3. Divide x^{3} – 5x^{2} + 8x – 4 by (x -2)

Drafted by Eunice (Maths)