Problem 22 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(2 x^{5}-1\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form of \( (2x^{5}-1)^{4} \) using the binomial theorem is \(16x^{20} - 32x^{15} + 24x^{10} - 8x^{5} + 1\).

Step 1: Identify the components of the binomial

In the term \( (2x^{5}-1)^{4} \), \(a = 2x^{5}\), \(b = -1\) and \(n = 4\). These values will be used in the binomial theorem formula.

Step 2: Use the binomial theorem to expand the binomial

Now proceed to the formula for the binomial theorem: \( (a+b)^{n} = Σ \binom{n}{k} a^{n-k} b^{k} \) for k=0 to n. After substitution, we get \( (2x^{5}-1)^{4} = Σ \binom{4}{k} (2x^{5})^{4-k} (-1)^{k} \) for k=0 to 4

Step 3: Calculate each term of the result

Calculate the terms of the result, remembering that the binomial coefficient \( \binom{n}{k} \) is calculated as \( \frac{n!}{k!(n-k)!} \). Using these formulas, we get the terms: \n- When k=0, term = \( \binom{4}{0} (2x^{5})^{4} (-1)^{0} = 16x^{20} \)\n- When k=1, term = \( \binom{4}{1} (2x^{5})^{3} (-1)^{1} = -32x^{15} \)\n- When k=2, term = \( \binom{4}{2} (2x^{5})^{2} (-1)^{2} = 24x^{10} \)\n- When k=3, term = \( \binom{4}{3} (2x^{5})^{1} (-1)^{3} = -8x^{5} \)\n- When k=4, term = \( \binom{4}{4} (2x^{5})^{0} (-1)^{4} = 1 \)

Step 4: Write down the final result

By adding up all terms computed above, we obtain the expanded expression: \(16x^{20} - 32x^{15} + 24x^{10} - 8x^{5} + 1\)