Question Video: Expanding and Simplifying the Product of a Binomial and the Square of a Binomial Mathematics

Video Transcript

Simplify the algebraic expression 𝑥 minus two times 𝑥 plus six squared.

We are asked to expand and simplify the product of a binomial expression with the square of a binomial expression. To do this, we first note that multiplication is associative. Thus, we can evaluate the product of these binomials by first multiplying two of the binomials together. Let’s start by expanding the square of a binomial.

We can square 𝑥 plus six using the horizontal method. We begin by writing 𝑥 plus six squared in its expanded form, 𝑥 plus six times 𝑥 plus six. We can then distribute 𝑥 plus six over 𝑥 plus six as follows. So, we have 𝑥 times 𝑥 plus six plus six times 𝑥 plus six.

In each term, we have a product of a monomial and a binomial. We can expand each term by distributing the monomial factor over the binomial to get 𝑥 times 𝑥 plus 𝑥 times six plus six times 𝑥 plus six times six. Now, we simplify each product, then combine the like terms to get the trinomial 𝑥 squared plus 12𝑥 plus 36. Thus, 𝑥 minus two times 𝑥 plus six squared is equal to 𝑥 minus two times 𝑥 squared plus 12𝑥 plus 36.

We can now distribute the factor 𝑥 minus two over the trinomial. In each term, we have the product of a monomial and a binomial. We can expand each term by distributing the monomial factor over the binomial. We do this by recalling the product rule for exponents. This rule tells us that 𝑥 to the power of 𝑚 times 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of 𝑚 plus 𝑛.

And remember, whenever we’re talking about monomials, we’re talking about our powers being nonnegative integer values. So, in particular, the product rule for monomials only counts when 𝑚 and 𝑛 are nonnegative integers, although this formula is true for other values of 𝑚 and 𝑛.

We continue by evaluating the product for each term separately. Distributing 𝑥 squared gives 𝑥 cubed minus two 𝑥 squared. Distributing 12𝑥 gives 12𝑥 squared minus 24𝑥. And distributing 36 gives 36𝑥 minus 72. Finally, we combine like terms as follows, which simplifies to 𝑥 cubed plus 10𝑥 squared plus 12𝑥 minus 72.

In conclusion, we have shown that the algebraic expression 𝑥 minus two times 𝑥 plus six squared simplifies to 𝑥 cubed plus 10𝑥 squared plus 12𝑥 minus 72.