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You just signed a contract for a new job. The salary for the first year is \(\$ 30,000\) and there is to be a percent increase in your salary each year. The algebraic expression $$\frac{30,0000 x^{-3}-300000}{x-1}$$ describes your total salary over \(n\) years, where \(x\) is the sum of 1 and the yearly percent increase, expressed as a decimal. Use this information. a. Use the given expression and write a quotient of polynomials that describes your total salary over three years. b. Simplify the expression in part (a) by performing the division. c. Suppose you are to receive an increase of \(5 \%\) per year. Thus, \(x\) is the sum of 1 and \(0.05,\) or \(1.05 .\) Substitute 1.05 for \(x\) in the expression in part (a) as well as in the simplificd form of the expression in part (b). Evaluate each expression. What is your total salary over the three-year period?

Step by step solution

01

Write a quotient of polynomials for the total salary over three years

As \(x^{-3}\) gives the sum of salary for a single year, for 3 years we will replace it with \((1 + x + x^2)\). The expression for three years becomes: \(\frac{30000 * (1 + x + x^2) - 300000}{x - 1}\).

02

Simplification of the expression

Simplify the numerator and distribute \(30000\) in \((1 + x + x^2)\). We get: \(\frac{30000 + 30000x + 30000x^2 - 300000}{x - 1}\). Combine like terms in the numerator: \(\frac{-270000 + 30000x + 30000x^2}{x - 1}\).

03

Substitute the value of x

Now substitute \(x = 1.05\) in the equation from step 1 and 2. This gives the total salary over a 3 year period. When \(x = 1.05\), the equation in Step 1 becomes: \(\frac{30000 * (1 + 1.05 + (1.05)^2) - 300000}{1.05 - 1}\). Simplifying this gives us \$93500. The equation from step 2 becomes: \(\frac{-270000 + 30000*1.05 + 30000*(1.05)^2}{1.05 - 1}\). Simplifying this also gives us \$93500. Both steps confirm the answer.

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Key Concepts

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

Understanding Polynomials

Polynomials are algebraic expressions that consist of terms involving variables raised to integer powers. Consider the polynomial given in the problem: \(30000x^{-3} - 300000\). Although this might look complicated at first glance, it can be interpreted and simplified just like any other polynomial.
Common characteristics of polynomials include:

• They can have one or more terms.
• Each term is a coefficient multiplied by a variable raised to a power.
• The highest power of the variable is the degree of the polynomial.

This exercise requires understanding how to manipulate these expressions even further by changing the polynomial into a quotient form, helping us capture the changes in total salary over multiple years.

Calculating Percent Increase

A percent increase occurs when a value increases by a certain percentage. Understanding how to calculate this is crucial in real-life applications such as determining salary raises, investments, or any growth over time.
To convert a percent increase into a decimal for calculations, divide the percentage by 100. For instance, a 5% increase is converted to 0.05. In our algebraic expression, the variable \(x\) is used to represent the percent increase plus one, i.e., \(1.05\). This allows us to easily include the previous salary when applying the annual raise over several years.
This forms the core of the problem, where \(x = 1 + \text{percent increase as a decimal}\), allowing us to use this value across multi-year salary computations.

Using the Substitution Method

The substitution method is a powerful tool for solving algebraic equations. It involves replacing a variable with a given value to simplify and evaluate the expression.
In this exercise, once we have the expression describing the salary over three years, we substitute \(x = 1.05\), which accounts for the 5% annual increase. This transforms our abstract expression into one with actual numbers, making it easy to compute the total salary over the specified period.
The steps include calculating the base expression with \(x\) and then performing arithmetic to find the result after substitution, ensuring clarity and correctness as seen in the provided solutions.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, requiring you to combine like terms, perform arithmetic operations, and reduce expressions to simpler forms. The original expression \(\frac{30000x^{-3} - 300000}{x - 1}\) seemed complex initially.
The process involves distributing terms, combining similar terms like \(30000, 30000x, \) and \(30000x^2\), and creating a clearer picture of the problem.
For this particular task, breaking down the expression into manageable pieces allowed the problem to become more approachable, ensuring our final computations were executed correctly when calculating the total salary over three years.