Question

find the following product using the appropriate special product.

\\((x^3 - 15)^2\\)

Explanation:

Identify the appropriate special product formula

We need to find the product of the expression \((x^3 - 15)^2\). This expression is in the form of the Square of a Binomial Difference, which is a type of Special Products. The general formula is:

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

Define the terms for substitution

By comparing \((x^3 - 15)^2\) to the formula \((a - b)^2\), we identify the terms:

$$ a = x^3 $$

$$ b = 15 $$

Apply the algebraic formula

We substitute our defined terms into the expanded form \(a^2 - 2ab + b^2\):

$$ (x^3 - 15)^2 = (x^3)^2 - 2(x^3)(15) + (15)^2 $$

Simplify each term

Now we simplify each part of the expression:

• For the first term, apply the power of a power rule \((x^m)^n = x^{m \cdot n}\):

$$ (x^3)^2 = x^{3 \cdot 2} = x^6 $$

• For the middle term, multiply the coefficients:

$$ -2(x^3)(15) = -30x^3 $$

• For the last term, calculate the square of the constant:

$$ 15^2 = 225 $$

Combine the simplified terms

Combining all the simplified parts gives the final polynomial:

$$ x^6 - 30x^3 + 225 $$

Answer:

\(x^6 - 30x^3 + 225\)