Question

drag the operator to the correct location on the image.

which operation results in a binomial?

\\((3y^6 + 4) \quad \boxed{\phantom{x}} \quad (9y^{12} - 12y^6 + 16)\\)

\\(\cdot\\) \\(-\\) \\(+\\)

Explanation:

Identify the structure of the expressions

We are given two polynomial expressions:

$$ A = 3y^6 + 4 $$

and

$$ B = 9y^{12} - 12y^6 + 16 $$

We need to find which operator (\(\cdot\), \(-\), or \(+\)) placed between them results in a binomial (a polynomial with exactly two terms).

Analyze the multiplication operator

Let's test the multiplication operator (\(\cdot\)):

$$ (3y^6 + 4) \cdot (9y^{12} - 12y^6 + 16) $$

Notice that the terms fit the sum of cubes factorization pattern:

$$ (a + b)(a^2 - ab + b^2) = a^3 + b^3 $$

Let \(a = 3y^6\) and \(b = 4\).
Then:

$$ a^2 = (3y^6)^2 = 9y^{12} $$

$$ ab = (3y^6)(4) = 12y^6 $$

$$ b^2 = 4^2 = 16 $$

Thus, the expression matches the pattern perfectly:

$$ (3y^6 + 4)((3y^6)^2 - (3y^6)(4) + 4^2) $$

Simplify the product

Using the sum of cubes formula:

$$ (3y^6 + 4)(9y^{12} - 12y^6 + 16) = (3y^6)^3 + 4^3 $$

$$ = 27y^{18} + 64 $$

The resulting polynomial \(27y^{18} + 64\) has exactly two terms, which makes it a binomial.

Evaluate other operators

Let's check addition (\(+\)):

$$ (3y^6 + 4) + (9y^{12} - 12y^6 + 16) = 9y^{12} - 9y^6 + 20 $$

This is a trinomial (three terms).

Let's check subtraction (\(-\)):

$$ (3y^6 + 4) - (9y^{12} - 12y^6 + 16) = -9y^{12} + 15y^6 - 12 $$

This is also a trinomial (three terms).

Therefore, the multiplication operator (\(\cdot\)) is the correct choice.

Answer:

Which operation results in a binomial?

$$(3y^6 + 4)$$

<blank>\(\cdot\)</blank>

$$(9y^{12} - 12y^6 + 16)$$