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Chapter 5: Problem 102

Perform the indicated operations. $$ (3 a-2 b)^{3} $$

Short Answer

Expert verified
The expanded expression is \(27a^3 - 54a^2b + 36ab^2 - 8b^3\).

Step by step solution

01

Expand the Expression

To expand \((3a - 2b)^3\), we will apply the binomial theorem. According to the binomial theorem, \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). In this case, \(x = 3a\), \(y = -2b\), and \(n = 3\).
02

Apply Binomial Coefficients

The binomial coefficients for cubic terms are \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\). These coefficients are 1, 3, 3, and 1, respectively. We will use these to expand the expression.
03

Substitute and Simplify Terms

Using the binomial coefficients, expand the terms:\[ (3a - 2b)^3 = \binom{3}{0}(3a)^3(-2b)^0 + \binom{3}{1}(3a)^2(-2b)^1 + \binom{3}{2}(3a)^1(-2b)^2 + \binom{3}{3}(3a)^0(-2b)^3 \]Calculating each term:\(= 1 imes (27a^3) imes 1 + 3 imes (9a^2) imes (-2b) + 3 imes (3a) imes (4b^2) + 1 imes 1 imes (-8b^3)\).
04

Calculate the Expression

Replace and simplify:\[ (3a - 2b)^3 = 27a^3 - 54a^2b + 36ab^2 - 8b^3 \].
05

Final Simplified Expression

Combine all the terms from the expanded expression to get the final result:\[ 27a^3 - 54a^2b + 36ab^2 - 8b^3 \] is the fully expanded result of \((3a - 2b)^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expansion
Polynomial expansion is a process of expressing a power of a binomial as a sum of terms involving powers of its components. Essentially, it's about decomposing a given expression, like \((3a - 2b)^3\), into its expanded form.Using formulas provided by the binomial theorem, we can expand polynomials systematically. This theorem states that \((x + y)^n\) can be expanded into a series:
  • Each term in this series has the form \(\binom{n}{k} x^{n-k} y^k\), where \(k\) varies from \(0\) to \(n\).
  • In our example, \((3a - 2b)^3\), the task is to determine each term in the expansion using the components \(3a\) and \(-2b\), with an exponent of 3.
By expanding polynomials, we gain deeper insight into their behavior and structure, which can be useful for calculus, algebra, and more advanced mathematical operations.
Cubic Expressions
Cubic expressions involve raising a binomial to the third power, which results in a polynomial of degree three. In this specific case of \((3a - 2b)^3\), our goal is to find the expanded polynomial.Cubic expressions always result in 4 terms. Here's why:
  • For a term like \((3a - 2b)^3\), the highest degree term is \(a^3\), which means this term will have three factors of \(a\).
  • The next terms decrease in the power of \(a\) while increasing in the power of \(b\), until reaching \(b^3\).
This results in a multi-term polynomial: \(27a^3 - 54a^2b + 36ab^2 - 8b^3\). Understanding cubic expressions and how they expand helps with algebraic manipulation and solving complex equations.
Binomial Coefficients
The binomial coefficients are key to expanding expressions such as \((3a - 2b)^3\). These coefficients determine the weight of each term in the expansion and are derived from Pascal's triangle.When expanding a cubic expression, the binomial coefficients are found using combinations \(\binom{n}{k}\):
  • \(\binom{3}{0} = 1\) is the coefficient for the first term.
  • \(\binom{3}{1} = 3\) applies to the second term.
  • \(\binom{3}{2} = 3\) is for the third term.
  • \(\binom{3}{3} = 1\) for the final term.
These coefficients guide the expansion, ensuring each term has the correct proportional influence in addition to the powers of \(x\) and \(y\). Recognizing these patterns can greatly simplify the process of polynomial expansion.

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Most popular questions from this chapter

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