1. The problem is to expand the expression $$(2x-1)^4$$ using the binomial theorem. 2. Recall the binomial theorem: $$(a-b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k$$ where $a=2x$, $b=1$, and $n=4$. 3. Calculate the binomial coefficients for $n=4$: $\binom{4}{0}=1$, $\binom{4}{1}=4$, $\binom{4}{2}=6$, $\binom{4}{3}=4$, $\binom{4}{4}=1$. 4. Expand each term: - $k=0$: $\binom{4}{0}(2x)^4 (-1)^0 = 1 \cdot (2x)^4 \cdot 1 = 16x^4$ - $k=1$: $\binom{4}{1}(2x)^3 (-1)^1 = 4 \cdot 8x^3 \cdot (-1) = -32x^3$ - $k=2$: $\binom{4}{2}(2x)^2 (-1)^2 = 6 \cdot 4x^2 \cdot 1 = 24x^2$ - $k=3$: $\binom{4}{3}(2x)^1 (-1)^3 = 4 \cdot 2x \cdot (-1) = -8x$ - $k=4$: $\binom{4}{4}(2x)^0 (-1)^4 = 1 \cdot 1 \cdot 1 = 1$ 5. Combine all terms: $$16x^4 - 32x^3 + 24x^2 - 8x + 1$$ 6. This is the expanded form of $$(2x-1)^4$$. Final Answer: $$16x^4 - 32x^3 + 24x^2 - 8x + 1$$