Factorize fully 54𝑥 cubed minus
In this question, we are given an
algebraic expression that we are asked to fully factor. To do this, we can start by noting
that there are two terms. And they are the product of
constants and variables raised to nonnegative integer exponents. So this is a binomial.
The first thing we should always
check is for any common factor amongst all of the terms. We see that there is no shared
variable among the terms. However, both terms share a factor
of two. Taking out this factor of two gives
us two multiplied by 27𝑥 cubed minus eight 𝑦 cubed. Since we need to fully factor the
expression, we should check if we can factor the binomial further. We see that both terms have the
variables raised to the power of three. This should remind us of the
difference between two cubes. In fact, we can see that this is a
difference between two cubes, since 27𝑥 cubed is equal to three 𝑥 all cubed and
eight 𝑦 cubed is equal to two 𝑦 all cubed.
Therefore, we can rewrite the
expression as two multiplied by three 𝑥 cubed minus two 𝑦 cubed. To factor the difference between
two cubes, we can recall that 𝑎 cubed minus 𝑏 cubed is equal to 𝑎 minus 𝑏
multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared. We can use this formula to factor
our difference between two cubes by substituting 𝑎 equals three 𝑥 and 𝑏 equals
two 𝑦 into the formula. We obtain two multiplied by three
𝑥 minus two 𝑦 multiplied by three 𝑥 squared plus three 𝑥 times two 𝑦 plus two
𝑦 squared. We can then simplify the expression
to get two times three 𝑥 minus two 𝑦 multiplied by nine 𝑥 squared plus six 𝑥𝑦
plus four 𝑦 squared.
We cannot factor the expression any
further since there are no common factors in the terms of each factor and they do
not resemble any expression that we know how to factor. So we have shown that 54𝑥 cubed
minus 16𝑦 cubed is equal to two times three 𝑥 minus two 𝑦 multiplied by nine 𝑥
squared plus six 𝑥𝑦 plus four 𝑦 squared.