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Use polynomial identities to multiply [tex]\left(5-4 x^3\right)\left(5+4 x^3\right)[/tex].

A. [tex]25-4 x^9[/tex]
B. [tex]25-40 x^3+16 x^6[/tex]
C. [tex]25-4 x^6[/tex]
D. [tex]25-16 x^6[/tex]

Answer :

To solve this problem, we'll use a well-known algebraic identity called the difference of squares. This identity states that:

[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]

Here, we have the expression [tex]\((5 - 4x^3)(5 + 4x^3)\)[/tex].

Let's identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

- [tex]\(a = 5\)[/tex]
- [tex]\(b = 4x^3\)[/tex]

Using the difference of squares identity, we can substitute into the formula:

1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 5^2 = 25
\][/tex]

2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (4x^3)^2 = 16x^6
\][/tex]

3. Substitute into the identity:
[tex]\[
(5 - 4x^3)(5 + 4x^3) = a^2 - b^2 = 25 - 16x^6
\][/tex]

So, the product simplifies to:

[tex]\[
25 - 16x^6
\][/tex]

The correct answer is:

D. [tex]\(25-16 x^6\)[/tex]

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