Question

Expand and simplify using the binomial theorem. a) \((a+3 b)^{3}\) b) \((3 a-2 b)^{5}\) c) \((2 x-5)^{4}\)

Short answer

(a) \((a + 3b)^3 = a^3 + 9a^2b + 27ab^2 + 27b^3\)(b) \((3a - 2b)^5 = 243a^5 - 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 - 32b^5\)(c) \((2x - 5)^4 = 16x^4 - 160x^3 + 600x^2 - 1000x + 625\)

Step-by-step solution

Write the binomial theorem formula

The binomial theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]where \[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]\is the binomial coefficient.

Apply the binomial theorem to part (a)

For \[(a + 3b)^3\]\, identify \[a\] = a, \[b\] = 3b, \[n\] = 3.\Using the binomial theorem, expand:\[(a + 3b)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k} (3b)^k\]Calculate each term individually:\[(a + 3b)^3 = \binom{3}{0} a^{3-0} (3b)^0 + \binom{3}{1} a^{3-1} (3b)^1 + \binom{3}{2} a^{3-2} (3b)^2 + \binom{3}{3} a^{3-3} (3b)^3\]Plug in binomial coefficients:\[(a + 3b)^3 = 1 \cdot a^3 + 3 \cdot a^2 \cdot 3b + 3 \cdot a \cdot (3b)^2 + 1 \cdot (3b)^3\]Simplify:\[(a + 3b)^3 = a^3 + 9a^2b + 27ab^2 + 27b^3\]

Apply the binomial theorem to part (b)

For \[(3a - 2b)^5\]\, identify \[a\] = 3a, \[b\] = -2b, \[n\] = 5.\Using the binomial theorem, expand:\[(3a - 2b)^5 = \sum_{k=0}^{5} \binom{5}{k} (3a)^{5-k} (-2b)^k\]Calculate each term individually:\[(3a - 2b)^5 = \binom{5}{0} (3a)^{5-0} (-2b)^0 + \binom{5}{1} (3a)^{5-1} (-2b)^1 + \binom{5}{2} (3a)^{5-2} (-2b)^2 + \binom{5}{3} (3a)^{5-3} (-2b)^3 + \binom{5}{4} (3a)^{5-4} (-2b)^4 + \binom{5}{5} (3a)^{5-5} (-2b)^5\]Plug in binomial coefficients:\[(3a - 2b)^5 = 1 (3a)^5 + 5 (3a)^4 (-2b) + 10 (3a)^3 (-2b)^2 + 10 (3a)^2 (-2b)^3 + 5 (3a)(-2b)^4 + 1 (-2b)^5\]Simplify:\[(3a - 2b)^5 = 243a^5 - 810a^4b + 1080a^3b^2 - 720a^2b^3 + 240ab^4 - 32b^5\]

Apply the binomial theorem to part (c)

For \[(2x - 5)^4\]\, identify \[a\] = 2x, \[b\] = -5, \[n\] = 4.\Using the binomial theorem, expand:\[(2x - 5)^4 = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} (-5)^k\]Calculate each term individually:\[(2x - 5)^4 = \binom{4}{0} (2x)^{4-0} (-5)^0 + \binom{4}{1} (2x)^{4-1} (-5)^1 + \binom{4}{2} (2x)^{4-2} (-5)^2 + \binom{4}{3} (2x)^{4-3} (-5)^3 + \binom{4}{4} (2x)^{4-4} (-5)^4\]Plug in binomial coefficients:\[(2x - 5)^4 = 1 (2x)^4 + 4 (2x)^3 (-5) + 6 (2x)^2 (-5)^2 + 4 (2x)(-5)^3 + 1 (-5)^4\]Simplify:\[(2x - 5)^4 = 16x^4 - 160x^3 + 600x^2 - 1000x + 625\]

Key concepts

Binomial Expansion

The binomial expansion using the Binomial Theorem is a way to expand expressions that are raised to any given power. This is particularly useful for simplifying and solving polynomial expressions. Using the Binomial Theorem, we can express \[(a + b)^n\] as the sum of terms in the form of \(\binom{n}{k} a^{n-k} b^k \), where \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\) is known as the binomial coefficient. These coefficients can be found using combinations, which shows how many ways you can choose \(k\) items from \(n\) items.

Polynomial Simplification

After applying the Binomial Theorem, we are often left with a series of terms that need to be simplified. Simplification involves combining like terms, reducing coefficients, and making the expression more concise. For instance, in the example of expanding \((a + 3b)^3\), we distributed and simplified the coefficients and exponents thoroughly. This led to the final, compact form of \(a^3 + 9a^2b + 27ab^2 + 27b^3\). The key steps in simplification include correctly applying the exponents, multiplying the coefficients, and combining like terms.

Combinatorics

Combinatorics is the field of mathematics that deals with counting combinations and permutations. In the context of the Binomial Theorem, combinatorics helps calculate the binomial coefficients. In our examples, this principle was used to expand expressions like \((2x - 5)^4 \). By calculating \( \binom{4}{k}\) for each term in the expansion sequence and simplifying, we obtained the expanded form of \((2x - 5)^4 = 16x^4 - 160x^3 + 600x^2 - 1000x + 625\). Understanding combinations is vital for binomial expansions and other advanced areas of algebra and statistics.