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Magazine Chapter 14: Problem 28
Use the binomial formula to expand each binomial. \((4-3 x)^{5}\)
Short Answer
Expert verified
\((4-3x)^5 = 1024 - 1920x + 8640x^2 - 17280x^3 + 12960x^4 - 243x^5\).
Step by step solution
01
Identify the Binomial Components
The given binomial expression is \((4 - 3x)^5\). Here, \(a = 4\) and \(b = -3x\). The exponent \(n = 5\).
02
Apply the Binomial Theorem
According to the Binomial Theorem, \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this problem, substitute \(a = 4\), \(b = -3x\), and \(n = 5\). The formula becomes \((4 - 3x)^5 = \sum_{k=0}^{5} \binom{5}{k} 4^{5-k} (-3x)^k\).
03
Calculate the Binomial Coefficients
Calculate \(\binom{5}{k}\) for \(k = 0, 1, 2, 3, 4, 5\). These are: \(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), \(\binom{5}{3} = 10\), \(\binom{5}{4} = 5\), \(\binom{5}{5} = 1\).
04
Expand Each Term in the Sum
Use previously calculated coefficients and expanding powers to write out the sum: \( \sum_{k=0}^{5} \binom{5}{k} 4^{5-k} (-3x)^k = \binom{5}{0} 4^5 (-3x)^0 + \binom{5}{1} 4^4 (-3x)^1 + \binom{5}{2} 4^3 (-3x)^2 + \binom{5}{3} 4^2 (-3x)^3 + \binom{5}{4} 4^1 (-3x)^4 + \binom{5}{5} 4^0 (-3x)^5 \).
05
Calculate Each Term
Calculate the value of each term:- For \(k = 0\), term is \(1 \cdot 4^5 \cdot 1 = 1024\).- For \(k = 1\), term is \(5 \cdot 4^4 \cdot (-3x) = -1920x\).- For \(k = 2\), term is \(10 \cdot 4^3 \cdot (9x^2) = 8640x^2\).- For \(k = 3\), term is \(10 \cdot 4^2 \cdot (-27x^3) = -17280x^3\).- For \(k = 4\), term is \(5 \cdot 4 \cdot 81x^4 = 12960x^4\).- For \(k = 5\), term is \(1 \cdot 1 \cdot (-243x^5) = -243x^5\).
06
Put It All Together
Combine all the calculated terms: \((4-3x)^5 = 1024 - 1920x + 8640x^2 - 17280x^3 + 12960x^4 - 243x^5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
The binomial expansion is a crucial mathematical concept used to express the power of a binomial as a sum of terms involving binomial coefficients. A binomial is an algebraic expression that contains exactly two different terms. For instance, \((a + b)^n\) represents a binomial raised to a positive integer power.
The Binomial Theorem provides a systematic way to expand such expressions:
The Binomial Theorem provides a systematic way to expand such expressions:
- Each term in the expansion is derived from raising the first term to decreasing powers and the second term to increasing powers.
- The process involves applying the formula \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where each term requires calculation of binomial coefficients.
Polynomial Expansion
Polynomial expansion involves expressing a polynomial, such as a binomial, in an expanded form. In the context of binomials, it provides an explicit breakdown of each term for powers of two items \( (a + b)^n \).
The process involves understanding the changes that occur in terms:
The process involves understanding the changes that occur in terms:
- We repeatedly multiply the terms in the binomial through successive terms until fully expanded.
- Each term of this expansion involves one of the terms in its original non-expanded polynomial raised to an increasingly higher power.
Binomial Coefficients
Binomial coefficients are the numerical factors that arise in the expansion of binomials. They reflect how the components of a binomial are combined in the expansion to form the polynomial terms.
They are denoted by \( \binom{n}{k} \), which signifies the coefficient of the term in which the binomial term \(b\) is raised to the power of \(k\):
They are denoted by \( \binom{n}{k} \), which signifies the coefficient of the term in which the binomial term \(b\) is raised to the power of \(k\):
- The notation \( \binom{n}{k} \) is read as "n choose k," representing the number of combinations possible when choosing \(k\) elements from \(n\) available.
- The coefficients can also be derived using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) denotes factorial.
