Question

Federal Pell Grants The amount (in dollars) of the average Pell Grant awarded in the years 1998 through 2005 can be approximated by \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\) where \(x\) represents the year, with \(x=8\) corresponding to 1998\. Evaluate the polynomial when \(x=14\) and \(x=15\). Then describe your results in everyday terms. (Source: U.S. Department of Education)

Short answer

Calculating the polynomial for \(x=14\) and \(x=15\) gives the average amount of Pell Grant for the years 2004 and 2005, respectively. The comparison of the two numbers provides insight into how the grant amount evolved between these years.

Step-by-step solution

Understanding the Polynomial

The given polynomial \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\) represents the amount of Pell Grant. Here, \(x=8\) represents the year 1998. Therefore, \(x=14\) represents 2004 and \(x=15\) represents 2005.

Evaluating the Polynomial for x=14

Substitute \(x=14\) in the polynomial \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\). Calculate the value to get the average amount of Pell grant for 2004.

Evaluating the Polynomial for x=15

Similarly, substitute \(x=15\) in the polynomial \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\). Calculate the value to get the average amount of Pell grant for 2005.

Interpreting the results

The calculated numbers represent the average Pell Grant for the years 2004 and 2005. Compare the values to understand how the grant amount changed over these years.

Key concepts

Federal Pell Grants

Federal Pell Grants are a form of financial aid provided by the U.S. Department of Education to students who need assistance to pay for college. These grants do not need to be repaid, which makes them a valuable resource for eligible students. The amount awarded can vary each year based on several factors, including federal funding levels and the student's financial need.

During the years 1998 through 2005, the average Pell Grant amounts were influenced by both economic conditions and federal budget priorities. A mathematical polynomial is used to approximate these amounts, providing a simplified model of otherwise complex financial distributions. By evaluating this polynomial, one can estimate the grant amount during specified years, offering a useful perspective on how support for students evolved in that period.

Polynomial Approximation

A polynomial approximation uses polynomial expressions to represent complex data or functions. These approximations help in simplifying real-world problems into more manageable mathematical terms.

The polynomial given for Pell Grants is a third degree, or cubic polynomial, which consists of terms like \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\). Such polynomials can approximate values across a range of inputs, where each term contributes differently based on the value of \(x\). The coefficients of each term dictate the impact of that particular power of \(x\) on the overall result.

Through polynomial approximation, one can model and predict trends, like estimating the average Pell Grant for a given year (e.g., 1998 through 2005). This technique turns historical data into a functional form, allowing educators and policymakers to visualize changes and possibly forecast future values.

Evaluating Polynomials

Evaluating a polynomial involves substituting a specific value into the polynomial to calculate a result. It's a straightforward process of plugging in the value of \(x\) and performing arithmetic to compute the final outcome.

For example, evaluating the Pell Grant polynomial for \(x=14\) and \(x=15\) means substituting these years' corresponding values into \(-4.874 x^{3}+155.85 x^{2}-1507.9 x+6443\). This will give us the average Pell Grant for the years 2004 and 2005.

Step-by-step, here's how it happens:

• Replace \(x\) with the specific year value.
• Calculate each term separately, ensuring you follow the order of operations (exponents first, then multiplication, and finally addition or subtraction).
• Add up all calculated terms to get the final value for the year.

By performing this evaluation, we can interpret how much average financial aid was available to students in those specific years, revealing the trends and shifts in educational support.