Problem 55 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((q-3)^{9} ;\) second term

Step-by-Step Solution

Answer

The second term of the binomial expansion \((q-3)^9\) is \(-27q^8\).

Step 1: Determine the values of a, b, n, and k

In this binomial expansion, the values are: a=q, b=-3, n=9, k=1 (since we want to find the second term)

Step 2: Calculate the binomial coefficient

Using the formula for the binomial coefficient \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) , let's calculate \(\binom{9}{1}\): \[\binom{9}{1} = \frac{9!}{1!(9-1)!} = \frac{9!}{1!8!}\] Since \(9! = 9 \times 8!\), we can simplify the expression: \[\frac{9!}{1!8!} = \frac{9 \times 8!}{1! \times 8!} = 9\]

Step 3: Substitute the values into the binomial theorem formula

Using the binomial theorem formula and the values we've determined, we have: \[\text{2nd term} = \binom{9}{1} \times q^{9-1} \times (-3)^1 = 9 \times q^8 \times (-3)\]

Step 4: Simplify to find the second term

Simplify the expression \[\text{2nd term} = 9 \times q^8 \times (-3) = -27q^8\] The second term of the binomial expansion \((q-3)^9\) is \(-27q^8\).