Problem 23 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (c+2)^{5} $$

Step-by-Step Solution

Answer

The simplified form of the expansion of \( (c+2)^{5} \) is \( c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32 \).

Step 1: Identify the Power

Here, the power to which the binomial is raised is 5. This means, according to the Binomial Theorem, that there will be 6 terms in the expanded expression (n + 1 terms, where n is the power).

Step 2: Apply the Binomial Theorem

Apply the Binomial Theorem to expand the binomial. The expansion of the binomial \( (c+2)^{5} \), according to the Binomial theorem, is given as: \( C(5,0)*c^{5}*2^{0} + C(5,1)*c^{4}*2^{1} + C(5,2)*c^{3}*2^{2} + C(5,3)*c^{2}*2^{3} + C(5,4)*c^{1}*2^{4} + C(5,5)*c^{0}*2^{5} \). Where C(n,k) denotes a combination, calculating the number of ways to choose k items from n.

Step 3: Calculate Combinations and Simplify

Next, calculate the value of the combinations and simplify the terms: \( 1*c^{5}*2^{0} + 5*c^{4}*2^{1} + 10*c^{3}*2^{2} + 10*c^{2}*2^{3} + 5*c^{1}*2^{4} + 1*c^{0}*2^{5} \), further simplifying to: \( c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32 \).