Factorise 80𝑥 squared minus five
To help us decide whether an
expression factorises into one or two brackets, we first see if there are any common
factors throughout the expression. If there are, that means we can
factorise into one bracket.
In this case, 80𝑥 squared and
negative five have a common factor of five. This means we can factorise this
expression into one bracket. And the highest common factor of
each term, five, goes on the outside of this bracket.
To find the terms inside the
bracket, we divide both of the terms in our original expression by five. 80 divided by five is 16, so 80𝑥
squared divided by five is 16𝑥 squared. Negative five divided by five is
negative one. The factorised form of 80𝑥 squared
minus five is, therefore, five multiplied by 16𝑥 squared minus one.
We aren’t finished yet though. Notice how the expression inside
the brackets has two square numbers with a minus sign between them. This is a special kind of
expression that we can call the difference of two squares.
To factorise an expression that’s
the difference of two squares, we can apply this formula. 𝑎 squared minus 𝑏 squared is
equal to 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏. Note that 𝑎 and 𝑏 can be any
algebraic expression, not just a list of square numbers that we’re familiar
Let’s take 16𝑥 squared minus
one. The square root of 16 is four, and
the square root of 𝑥 squared is simply 𝑥. That means that the term in the
front of each bracket is four 𝑥. The square root of one is one, so
the brackets become four 𝑥 plus one multiplied by four 𝑥 minus one.
We can use the FOIL method to
expand this back out to check. Four 𝑥 multiplied by four 𝑥 is
16𝑥 squared. Four 𝑥 multiplied by negative one
is negative four 𝑥. One multiplied by four 𝑥 is four
𝑥. And one multiplied by negative one
is negative one. Notice how the terms negative four
𝑥 and four 𝑥 cancel each other out. Negative four 𝑥 plus four 𝑥 is
zero. That leaves us with 16𝑥 squared
minus one, which is the expression we needed.
Now that we know we factorised the
expression 16𝑥 squared minus one correctly, we can pop it back into our earlier
expression of five lots of 16𝑥 squared minus one. That leaves us with five multiplied
by four 𝑥 plus one multiplied by four 𝑥 minus one.