Question

Which operation results in a binomial?

(3y^6+4) ? (9y^12-12y^6+16)
1.) +
2.) -
3.) .

Answer 1

3. · (times)

Explanation:

You want to know the operation that will give a binomial result from the expressions (3y^6 +4) and (9y^12 -12y^6 +16).

Like terms

The like terms in the two expressions are 3y^6 and -12y^6, or +4 and +16. Combining these terms by addition or subtraction will not result in the elimination of either the y^6 term or the constant term. Hence addition or subtraction cannot result in a binomial.

Sum of cubes

We know that the sum of cubes is factored into the product of a binomial and a trinomial:

a³ +b³ = (a +b)(a² -ab +b²)

Comparing this form to the given binomial and trinomial, we see that we can choose ...

a = 3y^6

b = 4

to make the given factors match exactly. That tells us their product will be a binomial sum:

(3y^6 +4) × (9y^12 -12y^6 +16) = (3y^6)^3 +(4)^3 = 27y^18 +64

The operation "times" (·) will result in a binomial.