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Chapter 14: Problem 29

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

Short Answer

Expert verified
The expanded form of \( (2a+b)^6 \) using the Binomial theorem is \( 64 a^{6} + 192 a^{5} b + 240 a^{4} b^{2} + 160 a^{3} b^{3} + 60 a^{2} b^{4} + 12 a b^5 + b^{6} \).

Step by step solution

01

Understand the Binomial Theorem

The Binomial Theorem states that for any real numbers a and b and for any natural number n: \[ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^{k} \]which means, the expansion consists of terms of 'a' decreasing in power from n to 0, and terms of 'b' increasing in power from 0 to n. The coefficient of each term in the expansion is the binomial coefficient: \[ \binom{n}{k}\]. Here n choose k represents the number of combinations of n things taken k at a time.
02

Apply the Binomial Theorem

Given binomial expression is \( (2a+b)^6 \). Applying the Binomial Theorem to it:\[(2a+b)^6 = \binom{6}{0} (2a)^{6-0} b^{0} + \binom{6}{1} (2a)^{6-1} b^{1} + \binom{6}{2} (2a)^{6-2} b^{2} + \binom{6}{3} (2a)^{6-3} b^{3} + \binom{6}{4} (2a)^{6-4} b^{4} + \binom{6}{5} (2a)^{6-5} b^{5} + \binom{6}{6} (2a)^{6-6} b^{6} \]
03

Simplify each term

Simplify each term by calculating the binomial coefficients and powers of a and b:\[= 64 a^{6} + 6 * 32 a^{5} b + 15 * 16 a^{4} b^{2} + 20 * 8 a^{3} b^{3} + 15 * 4 a^{2} b^{4} + 6 * 2 a b^5 + b^{6}\]\[= 64 a^{6} + 192 a^{5} b + 240 a^{4} b^{2} + 160 a^{3} b^{3} + 60 a^{2} b^{4} + 12 a b^5 + b^{6}\]

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