Problem 56 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((u-2)^{7} ;\) fourth term

Step-by-Step Solution

Answer

The fourth term of the binomial expansion \((u-2)^7\) is \(-280u^4\).

Step 1: Determine n, k, a, and b

Given that the expression is \((u-2)^7\), we have: n = 7 (exponent), k = 4 (position of the term), a = u (first element), and b = -2 (second element).

Step 2: Apply the binomial theorem formula

Now we apply the binomial theorem formula: $$\text{term}_k = \binom{n}{k - 1} a^{n - (k - 1)} b^{k - 1}$$ We have already determined n, k, a, and b. Now we plug them into the formula: $$\text{term}_4 = \binom{7}{4 - 1} u^{7 - (4 - 1)} (-2)^{4 - 1}$$

Step 3: Calculate the term

Perform calculations: $$\text{term}_4 = \binom{7}{3} u^{7 - 3} (-2)^{4 - 1}$$ $$\text{term}_4 = \binom{7}{3} u^{4} (-2)^{3}$$ Now, simplifying using binomial coefficient and solving for the powers of u and -2: $$\text{term}_4 = 35 u^4 (-8)$$ Final step, multiply the binomial coefficient by the powers: $$\text{term}_4 = -280 u^4$$ The fourth term of the binomial expansion \((u-2)^7\) is \(-280u^4\).