Question

find the following product using the appropriate special product.

\\((x^3 - 5)^2\\)

Explanation:

Identify the appropriate special product formula

We need to find the product of the expression \((x^3 - 5)^2\). This expression is in the form of a binomial squared, specifically a difference of two terms squared.

Using the Square of a Binomial Difference knowledge point, we apply the algebraic identity:

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

Define the terms for substitution

To apply the formula to our expression \((x^3 - 5)^2\), we identify the corresponding parts for \(a\) and \(b\):

• Let \(a = x^3\)
• Let \(b = 5\)

Substitute the terms into the formula

Using the Special Products knowledge point, we substitute our values of \(a\) and \(b\) into the expanded identity:

$$(x^3 - 5)^2 = (x^3)^2 - 2(x^3)(5) + (5)^2$$

Simplify each term of the expansion

Now we simplify each part of the expression:

• For the first term, apply the power of a power rule: \((x^3)^2 = x^{3 \cdot 2} = x^6\)
• For the middle term, multiply the coefficients: \(-2(x^3)(5) = -10x^3\)
• For the last term, square the constant: \(5^2 = 25\)

Combining these simplified terms gives:

$$x^6 - 10x^3 + 25$$

Answer:

Find the following product using the appropriate special product.

\((x^3 - 5)^2 =\) <blank>\(x^6 - 10x^3 + 25\)</blank>