Video Transcript
Factorize fully 𝑥 squared times 𝑦
minus five minus nine 𝑥 times 𝑦 minus five plus 18𝑦 minus 90.
It appears that a common factor 𝑥
squared has been pulled from the first two terms, revealing 𝑦 minus five in
parentheses. And a common factor of negative
nine 𝑥 has been pulled from the second pair of terms, revealing another 𝑦 minus
five in parentheses. In other words, prior to factoring
out 𝑥 squared, the first two terms must have been 𝑥 squared 𝑦 minus five 𝑥
squared. And prior to factoring out negative
nine 𝑥, the second pair of terms must have been negative nine 𝑥𝑦 plus 45𝑥.
Here, we can see the given
expression written as a polynomial with six terms. The six terms do not have a common
factor. However, we have shown that the
first pair of terms share a common factor 𝑥 squared and the second pair of terms
share a different common factor negative nine 𝑥. And we also know that the pairs of
factored terms share a common binomial factor of 𝑦 minus five. This leads us to wonder whether
this expression can be factored by grouping.
To determine this, we need to find
a common factor of the last two terms. Thankfully, 18𝑦 and negative 90 do
share a common factor. And that’s positive 18. When each term has a factor of 18
taken out, we have 𝑦 minus five in parentheses. And this matches the common factor
revealed in the first two pairs of factored terms. This is exactly what we expect to
see when factoring by grouping.
In this method, the last step is to
factor the expression by identifying a common factor in the factored terms. In this case, that is 𝑦 minus
five. The other factor consists of terms
made of the three common factors pulled from each pair of terms, shown here in
pink. We are asked to factorize fully, so
we must check to see if the quadratic factor is prime or can still be factored.
To factor 𝑥 squared minus nine 𝑥
plus 18, we must find two integers that have a product of 18 and a sum of negative
nine. Two such numbers exist. They are negative six and negative
three. So 𝑥 squared minus nine 𝑥 plus 18
is not prime, because the factors are 𝑥 minus six and 𝑥 minus three. Altogether, the full factorization
of the given expression is 𝑦 minus five times 𝑥 minus six times 𝑥 minus
three.
