Question

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

\\((x^3 + 10)(x^3 + 10)\\)

Explanation:

Identify the special product pattern

The given expression is:

$$ (x^3 + 10)(x^3 + 10) $$

This represents a binomial multiplied by itself, which is the square of a binomial.
Using the Square of a Binomial Sum knowledge point, we can rewrite this as:

$$ (x^3 + 10)^2 $$

Apply the binomial square formula

Using the Special Products formula for the square of a sum:

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

Here, we define our terms:

• \( a = x^3 \)
• \( b = 10 \)

Expand each term of the formula

We calculate each part of the expanded formula:

• First term squared:

$$ a^2 = (x^3)^2 = x^6 $$

• Twice the product of the terms:

$$ 2ab = 2(x^3)(10) = 20x^3 $$

• Second term squared:

$$ b^2 = 10^2 = 100 $$

Combine the expanded terms

We combine the terms to get the final simplified polynomial:

$$ x^6 + 20x^3 + 100 $$

Answer:

\(x^6 + 20x^3 + 100\)