Video Transcript
Factorize fully 49𝑎 squared minus 64𝑏 squared 𝑐 to the fourth power.
In this question, we are given an algebraic expression and asked to factor the
expression fully. To do this, we can start by noticing that every term in this expression is a
monomial, since the variables are raised to nonnegative integer exponents.
Since there are two terms, we can also say that this is a binomial. To factor a polynomial, we can start by checking if the terms share any common
factors. In this case, there are no common nontrivial factors among the terms. We can now check to see if the binomial is similar to any of the special binomials
that we already know how to factor. If we do this, then we can note that this binomial is similar to a difference between
squares. This tells us that 𝑥 squared minus 𝑦 squared is equal to 𝑥 plus 𝑦 times 𝑥 minus
𝑦.
To see why this result is applicable, we need to note that both terms are actually
squares. First, we can note that seven 𝑎 all squared is equal to 49𝑎 squared, since seven
times seven is 49 and 𝑎 times 𝑎 is 𝑎 squared. Next, we can note that 64𝑏 squared 𝑐 to the fourth power is equal to eight 𝑏𝑐
squared all squared, since eight times eight is 64, 𝑏 times 𝑏 is 𝑏 squared, and
𝑐 squared times 𝑐 squared is 𝑐 to the fourth power.
We are now ready to use this result to factor the expression. We can start by writing each term as a square. We have seven 𝑎 all squared minus eight 𝑏𝑐 squared all squared. Then, we use the difference of squares formula to factor, with 𝑥 equal to seven 𝑎
and 𝑦 equal to eight 𝑏𝑐 squared. This gives us seven 𝑎 plus eight 𝑏𝑐 squared multiplied by seven 𝑎 minus eight
𝑏𝑐 squared.
We cannot factor this any further, since the terms in each binomial factor do not
share a nontrivial common divisor.
