Problem 29 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (2 a+b)^{6} $$

Step-by-Step Solution

Answer

The expanded form of \((2a + b)^6\) is \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\).

Step 1: Understanding the Binomial Theorem

The Binomial Theorem tells us how to expand a binomial raised to a certain power. For any natural number \( n \), and any real numbers \( a \) and \( b \), \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k}b^{k}\).

Step 2: Application of the Binomial Theorem

Now we put \( a \) as \( 2a \), \( b \) as \( b \), and \( n \) as \( 6 \) into the formula given by the Binomial Theorem. Following the formula, we find the expansion of \((2a + b)^6\) to be \(\sum_{k=0}^{6} {6\choose k} (2a)^{6-k} b^{k}\).

Step 3: Calculation and Simplification

Applying the equation from step 2, it becomes: \({6\choose0} (2a)^6 b^0 + {6\choose1} (2a)^5 b^1 + {6\choose2} (2a)^4 b^2 + {6\choose3} (2a)^3 b^3 + {6\choose4} (2a)^2 b^4 + {6\choose5} (2a)^1 b^5 + {6\choose6} (2a)^0 b^6\). This simplifies to \(64a^6 + 192a^5b + 240a^4b^2 + 160a^3b^3 + 60a^2b^4 + 12ab^5 + b^6\).