Factorize fully 64𝑦𝑥 squared
minus 64𝑦𝑥 plus 16𝑦.
In this question, we are given an
algebraic expression and asked to factor this expression fully. We can start by analyzing the
algebraic expression that we are given. We can note that there are three
terms. And in every term, the variables
are raised to nonnegative integer exponents, so this is a trinomial.
To factor any polynomial, we can
start by checking for any common factors among all of its terms since we can take
out any factor shared by all of the terms. In this case, we can see that the
coefficients of all three terms share a factor of 16. We should not stop here; we should
also check for any shared factors of the variables among the three terms. We note that all three terms share
a factor of 𝑦. However, if we check for a shared
factor of 𝑥, we can note that the final term does not have a factor of 𝑥, so we
cannot factor this out of all three terms.
We call 16𝑦 the greatest common
divisor of these three monomial terms. Taking out the shared factor of
16𝑦 from the three terms yields 16𝑦 times four 𝑥 squared minus four 𝑥 plus
one. Since we need to factor the
expression fully, we need to see if we can further factor this new trinomial. To do this, we can start by
checking if the trinomial resembles any special trinomial we know how to factor.
We can note that both the first and
last terms are squares, since two 𝑥 all squared is four 𝑥 squared and one squared
equals one. Since the middle term is negative,
we can compare the expression to a perfect square of the form 𝑎 minus 𝑏 all
squared equals 𝑎 squared minus two 𝑎𝑏 plus 𝑏 squared. We can rewrite this trinomial in
this form by noting that negative four 𝑥 is equal to negative two times two 𝑥
We can now see that this is in the
form of a formula for a perfect square, with 𝑎 equal to two 𝑥 and 𝑏 equal to
one. Substituting 𝑎 equals two 𝑥 and
𝑏 equals one into this formula allows us to factor the expression to obtain 16𝑦
times two 𝑥 minus one squared. We cannot factor this expression
any further, since the two binomial terms do not share any nontrivial factor. Hence, we have fully factored the
expression to get 16𝑦 times two 𝑥 minus one squared.