Video Transcript
Expand the expression 𝑥 plus two
𝑦 multiplied by 𝑥 squared minus four 𝑥𝑦 plus three 𝑦 squared, giving your
answer in its simplest form.
The expression we’ve been asked to
expand is the product of a binomial, 𝑥 plus two 𝑦, and a trinomial, 𝑥 squared
minus four 𝑥𝑦 plus three 𝑦 squared. To answer this question, we need to
recall the distributive property of multiplication over addition. This states that 𝑎 multiplied by
𝑏 plus 𝑐 is equal to 𝑎𝑏 plus 𝑎𝑐. In other words, we can distribute
the multiplication by 𝑎 over the sum.
We could distribute the binomial
over the trinomial, but it will be simpler to do it the other way round. We may therefore find it helpful to
write the factors in the opposite order. Applying the distributive property
with the trinomial taking the role of 𝑎, 𝑥 taking the role of 𝑏, and two 𝑦
taking the role of 𝑐 gives 𝑥 multiplied by 𝑥 squared minus four 𝑥𝑦 plus three
𝑦 squared plus two 𝑦 multiplied by 𝑥 squared minus four 𝑥𝑦 plus three 𝑦
squared.
Each part of the expression on the
right-hand side is now the product of a monomial and a trinomial. And so we can apply the
distributive property again, this time distributing each monomial over the
trinomial. Distributing 𝑥 over the trinomial
— and recalling that when we multiply powers of the same base, we add the exponents
— gives 𝑥 cubed minus four 𝑥 squared 𝑦 plus three 𝑥𝑦 squared. And then distributing two 𝑦 over
the trinomial gives two 𝑥 squared 𝑦 minus eight 𝑥𝑦 squared plus six 𝑦
cubed.
Finally, we simplify the expression
by combining like terms, which are those that have the same powers of both 𝑥 and
𝑦. This gives 𝑥 cubed minus two 𝑥
squared 𝑦 minus five 𝑥𝑦 squared plus six 𝑦 cubed. So, by applying the distributive
property twice, we’ve expanded the given expression and found its simplest form: 𝑥
cubed minus two 𝑥 squared 𝑦 minus five 𝑥𝑦 squared plus six 𝑦 cubed.