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Chapter 4: Problem 89

Find each product. $$ (5 r-s)^{3} $$

Short Answer

Expert verified
125r^3 - 75r^2s + 15rs^2 - s^3

Step by step solution

01

Identify the binomial

Recognize that the given expression \(5r - s)^{3}\) is a binomial raised to a power. We will expand this binomial using the binomial theorem.
02

Recall the binomial theorem

The binomial theorem states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \]. In this case, \(a = 5r\), \(b = -s\), and \(n = 3\).
03

Write the binomial expansion

Using the binomial theorem formula for \( (5r - s)^3 \), we get:\[ (5r - s)^{3} = \sum_{k=0}^{3} \binom{3}{k} (5r)^{3-k} (-s)^k \]
04

Calculate each term in the expansion

Calculate each term individually:- For \(k = 0\): \binom{3}{0} (5r)^{3} (-s)^{0} = 1 \cdot (5r)^{3} \cdot 1 = 125r^{3}- For \(k = 1 \): \binom{3}{1} (5r)^{2} (-s) = 3 \cdot 25r^{2} \cdot (-s) = -75r^{2}s - For \(k = 2 \): \binom{3}{2} (5r) ( (-s)^{2} ) = 3 \cdot 5r \cdot s^{2} = 15rs^{2}- For \(k = 3 \): \binom{3}{3} (5r)^{0} ( (-s)^{3} ) = 1 \cdot 1 \cdot (-s)^{3} = -s^{3}\
05

Combine all the terms

Combine all the calculated terms together to get the expanded form of the expression: \[ (5r-s)^{3} = 125r^3 - 75r^2s + 15rs^2 - s^3 \]

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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 provides a formula to expand expressions that are raised to a power. This is particularly useful when dealing with binomials, which are algebraic expressions containing two terms. The theorem states:

Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. These expressions represent values and can be used to model real-world situations. An important aspect of algebraic expressions is recognizing their constituent parts:
  • Terms: Each individual part of the expression separated by a plus (+) or minus (-) sign.
  • Coefficients: The numerical part of a term that multiplies the variable(s).
  • Variables: Symbols, usually letters, that represent unknown values.
  • Exponents: The power to which a variable is raised.
In the provided exercise, the expression (5r - s) is a binomial because it has two terms, 5r and -s.

By using the binomial theorem, we can expand this binomial when it's raised to a power, such as in our case with (5r - s)^3.

The expanded form can reveal more about the behavior of the expression, especially how the variables and coefficients interact.


n. The binomial theorem states that
Polynomial Expansion
Polynomial expansion involves expressing a polynomial that has been raised to a power in its expanded form. This can be simplified using the binomial theorem:

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