Chapter 10: Problem 55
Use the Binomial Theorem to write the binomial expansion. \(\left(w^3-3\right)^4\)
Short Answer
Expert verified
The expansion of \((w^3-3)^4\) using the binomial theorem is \(w^{12} - 12w^{9} + 54w^6 - 108w^3 + 81\).
Step by step solution
01
Identifying the parts of the binomial expression
Identify the a, b, and n in the \((a + b)^n\) formula based on the binomial expression \((w^3-3)^4\). Here, \(a = w^3\), \(b = -3\), and \(n = 4\).
02
Write out the binomial expansion
Substitute \(a = w^3\), \(b = -3\), and \(n = 4\) into the binomial theorem formula. The binomial expansion will have \(n + 1 = 5\) terms. It becomes: \((w^3 - 3)^4 = \binom{4}{0}(w^3)^4*(-3)^0 +\binom{4}{1}(w^3)^3*(-3)^1 +\binom{4}{2}(w^3)^2*(-3)^2 +\binom{4}{3}(w^3)^1*(-3)^3 +\binom{4}{4}(w^3)^0*(-3)^4\)
03
Simplify Each Term of the Expansion
Now we simplify each term by calculating the binomial coefficients \(\binom{n}{k}\) and simplifying the powers: \(= \binom{4}{0}(w^{3*4})(-3)^0 + \binom{4}{1}(w^{3*3})(-3)^1 + \binom{4}{2}(w^{3*2})(-3)^2 + \binom{4}{3}(w^{3*1})(-3)^3 + \binom{4}{4}(w^{3*0})(-3)^4 = \= w^{12} - 4w^{9}*3 + 6w^6*(9) - 4w^3*(27) + 81\)
04
Finalize the Simplification
Lastly, simplify the expression by performing the remaining multiplication operations:\(= w^{12} - 12w^{9} + 54w^6 - 108w^3 + 81\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
Binomial expansion is an essential concept in algebra that deals with raising a binomial expression to any power. The process of binomial expansion follows the Binomial Theorem, which provides a formula to expand expressions of the form \(a + b)^n\), where \(a\) and \(b\) are any numbers, and \(n\) is a non-negative integer.
Utilizing this theorem, the expansion of \(w^3-3)^4\) as shown in the exercise is systematically broken down into a series of terms. Each term is generated by multiplying the binomial coefficients with the corresponding power of \(a\) and \(b\). The coefficients themselves are derived from the formula \(\binom{n}{k}\), where \(k\) ranges from \(0\) to \(n\). These coefficients are a crucial part of the binomial expansion since they determine how many times each term will appear in the expansion.
To make binomial expansion more approachable, understanding the pattern and symmetry in the coefficients can be helpful, as they follow Pascal's Triangle. The actual expansion includes terms that progressively decrease the power of the first variable \(a\) by 1, and increase the power of the second variable \(b\) by 1, starting from \(b^0\) up to \(b^n\). Thus, the binomial expansion of \(w^3-3)^4\) consists of five terms, since there are four plus one terms in any binomial expansion raised to the fourth power.
Utilizing this theorem, the expansion of \(w^3-3)^4\) as shown in the exercise is systematically broken down into a series of terms. Each term is generated by multiplying the binomial coefficients with the corresponding power of \(a\) and \(b\). The coefficients themselves are derived from the formula \(\binom{n}{k}\), where \(k\) ranges from \(0\) to \(n\). These coefficients are a crucial part of the binomial expansion since they determine how many times each term will appear in the expansion.
To make binomial expansion more approachable, understanding the pattern and symmetry in the coefficients can be helpful, as they follow Pascal's Triangle. The actual expansion includes terms that progressively decrease the power of the first variable \(a\) by 1, and increase the power of the second variable \(b\) by 1, starting from \(b^0\) up to \(b^n\). Thus, the binomial expansion of \(w^3-3)^4\) consists of five terms, since there are four plus one terms in any binomial expansion raised to the fourth power.
Binomial Coefficients
Binomial coefficients, often denoted as \(\binom{n}{k}\) or \(nCk\), play a pivotal role in the binomial expansion. They are the numbers that appear as coefficients in the binomial theorem. These coefficients represent the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order (combinations). They can be found in Pascal's Triangle or calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\), and it is the product of all positive integers less than or equal to \(n\).
In the context of our exercise, the coefficients \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\) are used to determine how many combinations of \(w^3\) and \( -3\) there are in each term of the expansion. These coefficients are calculated as 1, 4, 6, 4, and 1 respectively, and they are then multiplied by the corresponding terms of the expanded polynomial to arrive at the final expression. Understanding binomial coefficients is crucial for simplifying binomial expressions and solving combinatorial problems.
In the context of our exercise, the coefficients \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\) are used to determine how many combinations of \(w^3\) and \( -3\) there are in each term of the expansion. These coefficients are calculated as 1, 4, 6, 4, and 1 respectively, and they are then multiplied by the corresponding terms of the expanded polynomial to arrive at the final expression. Understanding binomial coefficients is crucial for simplifying binomial expressions and solving combinatorial problems.
Polynomial Simplification
The last stage in working with binomial expansions is the polynomial simplification process, which involves combining like terms and performing arithmetic to reach the simplest form of the expression. This process takes the expanded form, which consists of terms with binomial coefficients, variables raised to different powers, and constants, and transforms it into a more compact and readable polynomial.
In the solution provided, we start by calculating each term's binomial coefficient and powers of \(w^3\) and \( -3\). Afterward, we multiply these values to generate the terms of the expanded polynomial. It's important to note that here, the terms \( (-3)^0\), \( (-3)^1\), and so on result in alternating positive and negative signs, affecting the subsequent arithmetic. Once each term is calculated, we combine similar terms and reduce the expression to its simplest form, according to the laws of exponents and basic arithmetic operations.
For instance, in the simplified equation \(w^{12} - 12w^{9} + 54w^6 - 108w^3 + 81\), the like terms have been merged, and calculation of coefficients has been performed, resulting in a final polynomial that is easier to analyze or use for further calculations. Polynomial simplification requires careful attention to ensure all coefficients are correctly calculated and that the proper signs are applied throughout the process.
In the solution provided, we start by calculating each term's binomial coefficient and powers of \(w^3\) and \( -3\). Afterward, we multiply these values to generate the terms of the expanded polynomial. It's important to note that here, the terms \( (-3)^0\), \( (-3)^1\), and so on result in alternating positive and negative signs, affecting the subsequent arithmetic. Once each term is calculated, we combine similar terms and reduce the expression to its simplest form, according to the laws of exponents and basic arithmetic operations.
For instance, in the simplified equation \(w^{12} - 12w^{9} + 54w^6 - 108w^3 + 81\), the like terms have been merged, and calculation of coefficients has been performed, resulting in a final polynomial that is easier to analyze or use for further calculations. Polynomial simplification requires careful attention to ensure all coefficients are correctly calculated and that the proper signs are applied throughout the process.
