Question Video: Multiplying a Binomial by a Trinomial | Nagwa Question Video: Multiplying a Binomial by a Trinomial | Nagwa

Question Video: Multiplying a Binomial by a Trinomial Mathematics • First Year of Preparatory School

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Using the multiplication of a binomial by a trinomial, simplify the algebraic expression (3π‘₯ + 2𝑦)Β³.

02:27

Video Transcript

Using the multiplication of a binomial by a trinomial, simplify the algebraic expression three π‘₯ 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 π‘₯ plus two 𝑦 cubed equals three π‘₯ plus two 𝑦 times three π‘₯ plus two 𝑦 times three π‘₯ 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 π‘₯ times three π‘₯ is nine π‘₯ squared. Three π‘₯ times two 𝑦 is six π‘₯𝑦. Two 𝑦 times three π‘₯ is six π‘₯𝑦. And two 𝑦 times two 𝑦 is four 𝑦 squared. Adding these together, we get nine π‘₯ squared plus six π‘₯𝑦 plus six π‘₯𝑦, which is 12π‘₯𝑦, 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 π‘₯ squared times three π‘₯ is 27π‘₯ cubed. Nine π‘₯ squared times two 𝑦 is 18π‘₯ squared 𝑦. 12π‘₯𝑦 times three π‘₯ is 36π‘₯ squared 𝑦. 12π‘₯𝑦 times two 𝑦 is 24π‘₯𝑦 squared. Four 𝑦 squared times three π‘₯ is 12π‘₯𝑦 squared. And four 𝑦 squared times two 𝑦 is eight 𝑦 cubed.

Adding all this stuff together, we get 27π‘₯ cubed plus 18π‘₯ squared 𝑦 plus 36π‘₯ squared 𝑦 plus 24π‘₯𝑦 squared plus 12π‘₯𝑦 squared plus eight 𝑦 cubed. Finally, we collect like terms for 27π‘₯ cubed plus 54π‘₯ squared 𝑦 plus 36π‘₯𝑦 squared plus eight 𝑦 cubed.

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