The following expression is an odd
number when 𝑥 is an integer. 𝑥 squared multiplied by four 𝑥
minus one plus 𝑥 minus two all squared minus five. Prove this algebraically.
Before we start this question, it
is worth remembering that any expression that can be written in the form two
multiplied by 𝑛, where 𝑛 is an integer will represent an even number. This is because when we multiply an
integer by two, our answer is always even.
Every even number is preceded by an
odd number and it is also followed by an odd number. Therefore, any expression that can
be written in the form two 𝑛 plus one or two 𝑛 minus one must represent an odd
number, once again where 𝑛 is an integer.
Let’s now consider our
expression. Our first step is to expand the
brackets and simplify the expression. 𝑥 squared multiplied by four 𝑥 is
four 𝑥 cubed and 𝑥 squared multiplied by negative one is negative 𝑥 squared. In order to expand 𝑥 minus two all
squared, we need to write the bracket out twice and use the FOIL method.
Multiplying the first terms 𝑥
multiplied by 𝑥 gives us 𝑥 squared. Multiplying the outside terms gives
us negative two 𝑥 as 𝑥 multiplied by negative two is negative two 𝑥. Multiplying the inside two terms
also gives us negative two 𝑥. And finally, multiplying the last
terms gives us positive four. Simplifying this by grouping our
like terms gives us 𝑥 squared minus four 𝑥 plus four.
Our final step is to drop the minus
five into the next line of our working. Grouping the like terms allows us
to cancel the 𝑥 squareds as negative 𝑥 squared plus 𝑥 squared is equal to
zero. We can also group positive four and
negative five. This leaves us with four 𝑥 cubed
minus four 𝑥 minus one. The first two terms are divisible
by two. Therefore, we can factorize out a
two. This gives us two multiplied by two
𝑥 cubed minus two 𝑥 minus one.
We were told in the question that
𝑥 is an integer. And if 𝑥 is an integer, this means
that two 𝑥 cubed minus two 𝑥 will also be an integer. If we let two 𝑥 cubed minus two 𝑥
equal 𝑛, our expression can be written two 𝑛 minus one.
As the expression can be written in
the form two 𝑛 minus one, we can say that the expression is an odd number when 𝑥
is an integer.