Question

find the product of the binomials using the appropriate special product.

\\((x^3 + 2)(x^3 + 2)\\)

Explanation:

Identify the special product form

We are asked to find the product of the binomials:

$$ (x^3 + 2)(x^3 + 2) $$

This expression represents a binomial multiplied by itself, which can be written as a square:

$$ (x^3 + 2)^2 $$

This matches the Square of a Binomial Sum formula, which is one of our Special Products:

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

Define the terms for substitution

To apply the formula, we identify the corresponding terms \(a\) and \(b\) from our expression \((x^3 + 2)^2\):

• Let \(a = x^3\)
• Let \(b = 2\)

Apply the algebraic formula

Now, we substitute \(a = x^3\) and \(b = 2\) into the expanded form \(a^2 + 2ab + b^2\):

$$ (x^3 + 2)^2 = (x^3)^2 + 2(x^3)(2) + (2)^2 $$

Simplify each term

We simplify each part of the expanded expression:

• First term: Using exponent rules, \((x^3)^2 = x^{3 \cdot 2} = x^6\)
• Middle term: Multiplying the coefficients, \(2(x^3)(2) = 4x^3\)
• Last term: Evaluating the square, \(2^2 = 4\)

Combining these terms gives:

$$ x^6 + 4x^3 + 4 $$

Answer:

Find the product of the binomials using the appropriate special product.

$$ (x^3 + 2)(x^3 + 2) = $$

<blank>\(x^6 + 4x^3 + 4\)</blank>