Question Video: Multiplying a Multivariable Binomial and a Trinomial Mathematics

Video Transcript

Expand and simplify 𝑥 minus two 𝑦 plus three times two 𝑥 plus 𝑦.

To expand the product of two polynomials, we need to distribute one polynomial over each term in the other. So that means we could distribute the trinomial over the binomial or the binomial over the trinomial. In this case, it seems the first option is more straightforward. So, let’s start with distributing the trinomial over each term in the binomial as follows. We can then expand each of the two terms by distributing the first factor over each trinomial. For the first term, we have two 𝑥 times 𝑥 plus two 𝑥 times negative two 𝑦 plus two 𝑥 times three, which simplifies to two 𝑥 squared minus four 𝑥𝑦 plus six 𝑥. For the second term, we have 𝑦 times 𝑥 plus 𝑦 times negative two 𝑦 plus 𝑦 times three, which simplifies to 𝑥𝑦 minus two 𝑦 squared plus three 𝑦.

Now we can add the expressions together and collect like terms to simplify. We obtain two 𝑥 squared minus three 𝑥𝑦 minus two 𝑦 squared plus six 𝑥 plus three 𝑦. We have fully expanded and simplified the product of 𝑥 minus two 𝑦 plus three times two 𝑥 plus 𝑦.