Video Transcript
Using the multiplication of a
binomial by a trinomial, simplify the algebraic expression three 𝑥 plus two 𝑦 all
cubed.
It seems a little strange that we
are being asked to multiply a binomial by a trinomial when there do not appear to be
any trinomials present. Don’t worry though; one will appear
shortly. Let’s start by writing out the
three factors of our cube explicitly: three 𝑥 plus two 𝑦 cubed equals three 𝑥
plus two 𝑦 times three 𝑥 plus two 𝑦 times three 𝑥 plus two 𝑦.
The way to approach an algebraic
expansion problem such as this when there are more than two factors is to multiply
the factors together in pairs until everything is expanded. Because multiplication of
polynomials is associative, it does not matter if we multiply the first two factors
together first and leave the third for later or leave the first factor and multiply
the other two first. Let’s do this and multiply the last
two factors first.
We can do this using a grid to
multiply each term of one factor by each term of the other. Three 𝑥 times three 𝑥 is nine 𝑥
squared. Three 𝑥 times two 𝑦 is six
𝑥𝑦. Two 𝑦 times three 𝑥 is six
𝑥𝑦. And two 𝑦 times two 𝑦 is four 𝑦
squared. Adding these together, we get nine
𝑥 squared plus six 𝑥𝑦 plus six 𝑥𝑦, which is 12𝑥𝑦, plus four 𝑦 squared.
And now we have a binomial and a
trinomial, which we can multiply together, again using a grid to multiply each term
of the binomial by each term of the trinomial. Nine 𝑥 squared times three 𝑥 is
27𝑥 cubed. Nine 𝑥 squared times two 𝑦 is
18𝑥 squared 𝑦. 12𝑥𝑦 times three 𝑥 is 36𝑥
squared 𝑦. 12𝑥𝑦 times two 𝑦 is 24𝑥𝑦
squared. Four 𝑦 squared times three 𝑥 is
12𝑥𝑦 squared. And four 𝑦 squared times two 𝑦 is
eight 𝑦 cubed.
Adding all this stuff together, we
get 27𝑥 cubed plus 18𝑥 squared 𝑦 plus 36𝑥 squared 𝑦 plus 24𝑥𝑦 squared plus
12𝑥𝑦 squared plus eight 𝑦 cubed. Finally, we collect like terms for
27𝑥 cubed plus 54𝑥 squared 𝑦 plus 36𝑥𝑦 squared plus eight 𝑦 cubed.
