High School

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Expand and combine like terms:

[tex]\[(2x^3 + 5x)(2x^3 - 5x)\][/tex]

Options:

A. [tex]\(4x^6 - 25x^3\)[/tex]

B. [tex]\(2x^6 - 25x^2\)[/tex]

C. [tex]\(4x^5 - 25x^2\)[/tex]

D. [tex]\(4x^6 - 25x^2\)[/tex]

Answer :

To expand and combine like terms for the expression [tex]\((2x^3 + 5x)(2x^3 - 5x)\)[/tex], follow these steps:

1. Recognize the Expression Type:
This expression can be viewed as a difference of squares, which follows the formula: [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex].

2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
In this context:
[tex]\[
a = 2x^3 \quad \text{and} \quad b = 5x
\][/tex]

3. Apply the Difference of Squares Formula:
Using the formula, substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(2x^3 + 5x)(2x^3 - 5x) = (2x^3)^2 - (5x)^2
\][/tex]

4. Calculate [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- Calculate [tex]\((2x^3)^2\)[/tex]:
[tex]\[
(2x^3)^2 = 4x^6
\][/tex]
- Calculate [tex]\((5x)^2\)[/tex]:
[tex]\[
(5x)^2 = 25x^2
\][/tex]

5. Combine the Results:
Substitute these back into the formula to get:
[tex]\[
4x^6 - 25x^2
\][/tex]

So, the expanded and combined expression is:
[tex]\[
4x^6 - 25x^2
\][/tex]

This gives us the expanded form of [tex]\((2x^3 + 5x)(2x^3 - 5x)\)[/tex].

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