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Chapter 5: Problem 47

Find each product. \((r+5)^{3}\)

Short Answer

Expert verified
The expanded form is \( r^3 + 15r^2 + 75r + 125 \).

Step by step solution

01

- Understand the cubic expansion formula

To expand \( (r+5)^3 \), recognize that it can be expanded using the binomial theorem. Specifically, \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \). In this case, \( a = r \) and \( b = 5 \).
02

- Expand using the binomial theorem

Substitute the values into the binomial theorem: \( (r+5)^3 = r^3 + 3r^2(5) + 3r(5^2) + 5^3 \).
03

- Calculate each term separately

Compute each term: \( r^3 \), \( 3r^2(5) = 15r^2 \), \( 3r(25) = 75r \), and \( 5^3 = 125 \).
04

- Sum all the terms

Add the computed terms together: \( r^3 + 15r^2 + 75r + 125 \).

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Key Concepts

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

Cubic expansion
Cubic expansion is a fundamental concept in algebra that deals with raising a binomial expression to the power of three. When we have an expression like \(r+5\)^3, it means we are multiplying \(r+5\) by itself three times. This can be tedious to multiply directly, which is why we use the binomial theorem. The binomial theorem helps simplify and streamline the process by providing a formula for any binomial raised to a power. For a cubic expansion, the formula is \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \). Here, you can directly plug in the values of \(a\) and \(b\) to get the expanded form.
Algebraic expressions
An algebraic expression includes numbers, variables, and the arithmetic operations involving addition, subtraction, multiplication, and division. In our example of expanding \(r+5\)^3, \(r\) and \(5\) form the algebraic expression. Understanding algebraic expressions is critical because it forms the foundation for manipulating and simplifying more complex mathematical problems. Each term in the expanded form \((r^3, 15r^2, 75r, 125)\) is also an algebraic expression. We use these expressions to represent real-world quantities and solve equations.
Polynomial expansion
Polynomial expansion involves expressing a polynomial in a more extended form. When using the binomial theorem, we transform a compact polynomial, such as \(r+5\), into a sum of multiple terms. These terms make it easier to analyze and understand the polynomial's behavior. For example, expanding \(r+5\)^3 yields \(r^3 + 15r^2 + 75r + 125\). Each term represents a part of the whole polynomial, showing the contribution of different degrees of the variable \(r\). By mastering polynomial expansion, students can tackle a wide range of algebraic problems more effectively.

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Most popular questions from this chapter

Perform each division using the "long division" process. $$ \frac{5 z^{3}-z^{2}+10 z+2}{z+2} $$

Evaluate. $$ 38 \div 10 $$

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Find each product. \(-3 k\left(8 k^{2}-12 k+2\right)\)

To understand how the special product \((a+b)^{2}=a^{2}+2 a b+b^{2}\) can be applied to a purely numerical problem. Evaluate \(35^{2},\) using either traditional paper-and-pencil methods or a calculator.
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