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Magazine Chapter 13: Problem 22
Carry out the indicated expansions. $$\left(5 A-B^{2}\right)^{3}$$
Short Answer
Expert verified
The expansion is \(125A^3 - 75A^2B^2 + 15AB^4 - B^6\).
Step by step solution
01
Identify the Binomial and Its Components
The expression given is \((5A - B^2)^3\). This is a binomial expression in which \(5A\) is the first term and \(B^2\) is the second term. We need to expand it using the binomial theorem.
02
Apply the Binomial Theorem
The binomial theorem states that \((x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k\). In our case, \(x = 5A\) and \(y = -B^2\), and \(n = 3\).
03
Write Down the General Expansion Formula
Using the binomial theorem, the expansion of \((5A - B^2)^3\) becomes:\[(5A - B^2)^3 = \sum_{k=0}^{3} \binom{3}{k} (5A)^{3-k} (-B^2)^k\]
04
Compute Each Term in the Expansion
Compute each term of the expansion as follows:- For \(k = 0\): \(\binom{3}{0} (5A)^3 (-B^2)^0 = 125A^3\)- For \(k = 1\): \(\binom{3}{1} (5A)^2 (-B^2)^1 = 3 \times 25A^2 \times (-B^2) = -75A^2B^2\)- For \(k = 2\): \(\binom{3}{2} (5A)^1 (-B^2)^2 = 3 \times 5A \times B^4 = 15AB^4\)- For \(k = 3\): \(\binom{3}{3} (5A)^0 (-B^2)^3 = -B^6\)
05
Combine All the Terms
Now combine all terms from Step 4 to get:\[ 125A^3 - 75A^2B^2 + 15AB^4 - B^6\]
06
Finalize the Expression
Verify that each step applies correctly the binomial coefficients and powers. Write the final expanded form:\[ 125A^3 - 75A^2B^2 + 15AB^4 - B^6\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. Specifically, it's used for expanding expressions of the form \((x + y)^n\). This theorem is essential for breaking down complex binomial expressions into simpler terms that are easier to handle.
- The formula for the binomial theorem is: \((x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k\) where \(\binom{n}{k}\) represents the binomial coefficients.
- The binomial coefficients \(\binom{n}{k}\) indicate the number of ways to choose \(k\) items from \(n\) items, and they form the famous Pascal鈥檚 Triangle.
- This theorem can deal with both addition and subtraction; if we have subtraction, like \((x - y)^n\), the terms involving \(y\) would alternate in sign.
Polynomial Expansion
Polynomial expansion involves breaking down a polynomial expression into simpler, more manageable pieces. When we use a method like the binomial theorem to expand a polynomial, we essentially rewrite the expression as a sum of terms, each with coefficients and varying powers of the expression's components.
- Every term in a polynomial expansion is derived from picking a certain number of terms from the original expression and raising them to necessary powers.
- It is a vital technique in algebra because it transforms expressions, making them easier to analyze and solve.
- The expansion process involves systematically applying the chosen formula and carrying out operations to multiply and simplify.
Algebra
Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express relationships and patterns in a general way. Expanding a binomial expression using the binomial theorem is an excellent example of algebraic manipulation.
- Algebra involves working with variables and constants to form equations and expressions.
- Through the use of algebraic methods, like the binomial theorem, we convert complex problems into more manageable tasks.
- Understanding how to expand binomials is essential for solving higher-level math problems that involve relationships between variables.
