Answer :
Let's use polynomial identities to multiply [tex]\(\left(5-4x^3\right)\left(5+4x^3\right)\)[/tex].
This expression is of the form [tex]\((a - b)(a + b)\)[/tex], which is a difference of squares identity. The difference of squares identity states that:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
Here, we have:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 4x^3\)[/tex]
Using the difference of squares identity, we get:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
Plugging in [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(5 - 4x^3)(5 + 4x^3) = 5^2 - (4x^3)^2
\][/tex]
Now, calculate each term separately:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
(4x^3)^2 = 4^2 \cdot (x^3)^2 = 16x^6
\][/tex]
So, the expression simplifies to:
[tex]\[
25 - 16x^6
\][/tex]
Therefore, the result of multiplying [tex]\(\left(5-4x^3\right)\left(5+4x^3\right)\)[/tex] is:
[tex]\[
\boxed{25 - 16x^6}
\][/tex]
So, the correct answer is:
D. [tex]\(25 - 16 x^6\)[/tex]
This expression is of the form [tex]\((a - b)(a + b)\)[/tex], which is a difference of squares identity. The difference of squares identity states that:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
Here, we have:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 4x^3\)[/tex]
Using the difference of squares identity, we get:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
Plugging in [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(5 - 4x^3)(5 + 4x^3) = 5^2 - (4x^3)^2
\][/tex]
Now, calculate each term separately:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
(4x^3)^2 = 4^2 \cdot (x^3)^2 = 16x^6
\][/tex]
So, the expression simplifies to:
[tex]\[
25 - 16x^6
\][/tex]
Therefore, the result of multiplying [tex]\(\left(5-4x^3\right)\left(5+4x^3\right)\)[/tex] is:
[tex]\[
\boxed{25 - 16x^6}
\][/tex]
So, the correct answer is:
D. [tex]\(25 - 16 x^6\)[/tex]
