High School

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

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

Answer :

To solve the problem of multiplying [tex]\((5 - 4x^3)(5 + 4x^3)\)[/tex], we can use the identity known as the difference of squares. This identity tells us that:

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

In this case, we can identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:

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

Using the difference of squares identity, we substitute these values in:

[tex]\[
(5 - 4x^3)(5 + 4x^3) = 5^2 - (4x^3)^2
\][/tex]

Now let's calculate step-by-step:

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

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

Now, apply the identity:

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

The expression simplifies to the term:

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

So, the correct answer is:

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

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