Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Compound Interest After 3 years, an investment of \(\$ 1000\) earning an interest rate \(r\) compounded annually will be worth \(1000(1+r)^{3}\) dollars. Write this expression as a polynomial in standard form.

Step-by-Step Solution

Answer

The expression \(1000(1+r)^{3}\) in polynomial standard form is \(1000r^{3} + 3000r^{2} + 3000r + 1000\).

Step 1: Expand the binomial

First, expand the binomial \((1 + r)^3\). Using the binomial theorem, this is equal to \(1 + 3r + 3r^{2} + r^{3}\). This expansion uses the binomial coefficient regularly represented by 'n choose k', where n is the overall power and k is the specific term.

Step 2: Multiply by the coefficient

Now multiply each term of the expanded binomial by the constant, 1000. This results in \(1000 + 3000r + 3000r^{2} + 1000r^{3}\).

Step 3: Write the expression in standard form

The standard form of a polynomial orders the terms by descending power. So, the expression in a standard form would be \(1000r^{3} + 3000r^{2} + 3000r + 1000\).

Key Concepts

Binomial Theorem

In mathematics, the Binomial Theorem is a powerful tool that helps us expand expressions of the form \((a+b)^n\). It is particularly useful when dealing with powers larger than 2, making it a key component in algebra. For the expression \((1 + r)^3\), the Binomial Theorem allows us to expand this into a sum of terms. These terms include each power of \(r\) starting from 0 up to 3.

• The theorem uses binomial coefficients, often noted as 'n choose k' (written as \( \binom{n}{k} \)).
• For \((1 + r)^3\), you calculate these coefficients as \(\binom{3}{0}, \binom{3}{1}, \binom{3}{2},\) and \(\binom{3}{3}\) which are 1, 3, 3, and 1 respectively.
• The expansion results in \(1 + 3r + 3r^2 + r^3\).

Understanding binomial expansion is crucial in simplifying polynomial expressions and solving real-world problems, like calculating compound interest.

Polynomial Expansion

Polynomial expansion refers to expressing a power of a binomial as a sum of multiple terms. It's like pulling apart an expression to see how each part contributes to the whole. When you use the Binomial Theorem, you take a concise expression like \((1+r)^3\) and rewrite it in expanded form.

• This process involves using the calculated binomial coefficients to multiply each term that results from raising both \(1\) and \(r\) to a power.
• For \((1 + r)^3\), you distribute the coefficients over the powers resulting in \(1 + 3r + 3r^2 + r^3\).

Doing this kind of expansion is helpful because it simplifies calculations, especially when combined with other algebraic operations such as multiplication. Post-expansion, it is easier to apply additional calculations like multiplication by a constant.

Standard Form of a Polynomial

A polynomial is in standard form when its terms are written in descending order of their powers. This format is important as it creates a consistent way of writing and comparing polynomials. For our binomial expansion example, once we expanded \((1 + r)^3\), the terms were multiplied by 1000, yielding \(1000 + 3000r + 3000r^2 + 1000r^3\).

• Reordering this to descending power, we get: \(1000r^3 + 3000r^2 + 3000r + 1000\).
• This sequence from highest to lowest power makes evaluating them straightforward, especially when graphing or integrating such expressions.
• Standardizing the format also facilitates easier identification of key attributes of the polynomial, like its degree. 

Using standard form rules simplifies the process of analyzing polynomials and assists in identifying specific features quickly.