Which operation results in a binomial?
$\left(3 y^6+4
ight) \quad\left(9 y^{12}-12 y^6+16
ight)$
We are given two expressions: \(A = 3y^6 + 4\) and \(B = 9y^{12} - 12y^6 + 16\). Our goal is to identify which basic mathematical operation (addition, subtraction, multiplication, or division) results in a binomial, which is a polynomial consisting of exactly two terms. We will test each operation by simplifying the resulting expression.
First, we test addition and subtraction:
Neither of these operations results in a binomial.
Next, we test multiplication. We calculate the product \(A \times B = (3y^6 + 4)(9y^{12} - 12y^6 + 16)\).
Recognizing the algebraic pattern of the sum of cubes, let \(u = 3y^6\) and \(v = 4\).
Then \(u^2 = (3y^6)^2 = 9y^{12}\) and \(v^2 = 4^2 = 16\). The term \(uv = (3y^6)(4) = 12y^6\).
Thus, the product takes the form \((u + v)(u^2 - uv + v^2)\), which is a well-known identity for the sum of cubes: \(u^3 + v^3\).
Substituting back: \(A \times B = (3y^6)^3 + (4)^3 = 27y^{18} + 64\).
This expression has exactly two terms, making it a binomial.
Finally, we examine division. The division \(A / B = \frac{3y^6 + 4}{9y^{12} - 12y^6 + 16}\) does not simplify into a binomial. Since multiplication successfully results in a binomial, we have our answer. The operation that results in a binomial is multiplication. Thus, the result of the operation is \(\boxed{\text{multiplication}}\).
In fields like structural engineering or computer science, recognizing polynomial patterns is crucial for simplifying complex equations. For example, when calculating the stress on a beam or optimizing an algorithm's complexity, identifying that a product follows a specific identity like the sum of cubes can transform a long, multi-term expression into a concise two-term binomial. This makes further calculations, like finding roots or derivatives, much simpler and more efficient.