Fully factorise the expression eight 𝑝 squared minus 12𝑝 cubed.
To factorise an expression like this means we want to put it into brackets. We want to write it as a product of terms or expressions which multiply together to give the original expression. The word “Fully” is really important. It suggests that when we factorise, we will be able to take out more than one common factor. We need to make sure that we factorise by the highest common factor of the two terms.
Let’s begin by looking at the numbers in the two terms: they are eight and 12. With a little bit of thought, we see that the highest common factor or HCF for short of eight and 12 is four because four multiplied by two gives eight and four multiplied by three gives 12. So we can bring four out as a common factor.
Next, we look at the letters 𝑝 squared and 𝑝 cubed. 𝑝 squared means 𝑝 multiplied by 𝑝 and 𝑝 cubed means 𝑝 multiplied by 𝑝 multiplied by 𝑝 again. What these two terms have in common are two factors of 𝑝. In fact, 𝑝 cubed can be written as 𝑝 squared multiplied by 𝑝. So the highest common factor of 𝑝 squared and 𝑝 cubed is 𝑝 squared.
In fact, there’s a tip here. When we’re factorizing terms like this that’s powers of the same variable, the highest common factor which we can take out will be the lower power of that variable. So for 𝑝 squared and 𝑝 cubed, the lower power is two, which means the highest common factor we can take out is 𝑝 squared.
Next, we need to work out what goes inside the brackets, what we have to multiply four 𝑝 squared by to give the original expression. The first term is eight 𝑝 squared. So we only need to multiply four 𝑝 squared by two. Four multiplied by two is eight. So four 𝑝 squared multiplied by two is eight 𝑝 squared. There’s then a minus sign between the two terms and we need to make 12𝑝 cubed. Well, four multiplied by three gives 12. And as we’ve seen, 𝑝 squared multiplied by 𝑝 gives 𝑝 cubed. So the second term in the bracket that we need is three 𝑝 because four 𝑝 squared multiplied by three 𝑝 will give 12𝑝 cubed. We’ve got the negative sign between the terms already.
Now before we finish, we need to check that we have indeed fully factorised this expression. We need to check whether the terms inside the bracket have any further common factors. Well, two and three 𝑝 don’t have any common factors other than one, which means we can’t factorise this expression any further and so we’re done.
Now just suppose for a moment that you didn’t actually fully factorise this expressions straightaway. Suppose you didn’t realise that the highest common factor of eight and 12 was four. But instead, you took out a common factor of two. And suppose you didn’t realise that the highest common factor of 𝑝 squared and 𝑝 cubed was 𝑝 squared. And so, you only took out a factor of 𝑝. In this case, you’d have two 𝑝 outside the bracket and then four 𝑝 minus six 𝑝 squared inside the bracket so that when you expand, you get back to the original expression.
This would be a factorized form of eight 𝑝 squared minus 12𝑝 cubed. But it wouldn’t be fully factorised because if we look inside the bracket, we can see that there are still further common factors. Four and six have a common factor of two, which we can take outside the bracket. And 𝑝 and 𝑝 squared have a common factor of 𝑝. Inside the bracket, we then have two minus three 𝑝 because two times 𝑝 times two gives four 𝑝 and two times 𝑝 times negative three 𝑝 gives negative six 𝑝 squared. The factors outside the bracket can be simplified by multiplying. Two 𝑝 multiplied by two is four 𝑝 and then multiplying by 𝑝 gives four 𝑝 squared.
Notice we now have the same expression as we found during our first factorisation: four 𝑝 squared multiplied by two minus three 𝑝. What this means is that you don’t have to get to the fully factorized form in one step. It’s okay to take out common factors other than the highest common factor along the way. But you must keep checking whether or not there are any common factors left in your bracket in order to make sure that you fully factorised the expression.