Answer :
To multiply [tex]\((5 - 4x^3)(5 + 4x^3)\)[/tex], we can use a well-known polynomial identity called the difference of squares. This identity states:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In our problem, let's identify the parts that match this identity:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 4x^3\)[/tex]
According to the difference of squares identity, we need to square both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and then subtract the square of [tex]\(b\)[/tex] from the square of [tex]\(a\)[/tex].
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. Now, apply the difference of squares identity:
[tex]\[
(5 - 4x^3)(5 + 4x^3) = a^2 - b^2 = 25 - 16x^6
\][/tex]
Thus, the product of [tex]\((5 - 4x^3)(5 + 4x^3)\)[/tex] is:
[tex]\[
\boxed{25 - 16x^6}
\][/tex]
The correct answer is option D: [tex]\(25 - 16x^6\)[/tex].
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In our problem, let's identify the parts that match this identity:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 4x^3\)[/tex]
According to the difference of squares identity, we need to square both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and then subtract the square of [tex]\(b\)[/tex] from the square of [tex]\(a\)[/tex].
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. Now, apply the difference of squares identity:
[tex]\[
(5 - 4x^3)(5 + 4x^3) = a^2 - b^2 = 25 - 16x^6
\][/tex]
Thus, the product of [tex]\((5 - 4x^3)(5 + 4x^3)\)[/tex] is:
[tex]\[
\boxed{25 - 16x^6}
\][/tex]
The correct answer is option D: [tex]\(25 - 16x^6\)[/tex].
