To simplify the expression [tex]\(16x^4 - (x-3)^4\)[/tex], follow these steps:
1. Identify the components of the expression:
- The first term is [tex]\(16x^4\)[/tex].
- The second term is [tex]\((x-3)^4\)[/tex].
2. Expand [tex]\((x-3)^4\)[/tex]:
To expand [tex]\((x-3)^4\)[/tex], use the binomial theorem:
[tex]\[
(a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex]. Therefore,
[tex]\[
(x - 3)^4 = x^4 - 4x^3 \cdot 3 + 6x^2 \cdot 3^2 - 4x \cdot 3^3 + 3^4
\][/tex]
Simplify the terms:
[tex]\[
(x - 3)^4 = x^4 - 12x^3 + 54x^2 - 108x + 81
\][/tex]
3. Substitute the expanded form back into the original expression:
Substitute [tex]\((x - 3)^4\)[/tex] in the expression [tex]\(16x^4 - (x-3)^4\)[/tex]:
[tex]\[
16x^4 - (x^4 - 12x^3 + 54x^2 - 108x + 81)
\][/tex]
4. Distribute the negative sign:
[tex]\[
16x^4 - x^4 + 12x^3 - 54x^2 + 108x - 81
\][/tex]
5. Combine like terms:
[tex]\[
(16x^4 - x^4) + 12x^3 - 54x^2 + 108x - 81
\][/tex]
Simplify the first term:
[tex]\[
15x^4 + 12x^3 - 54x^2 + 108x - 81
\][/tex]
So, the simplified expression is:
[tex]\[
15x^4 + 12x^3 - 54x^2 + 108x - 81
\][/tex]
This is the final simplified form of the expression [tex]\(16x^4 - (x-3)^4\)[/tex].
