Problem 96 - College Algebra

Question

The polynomial \(8 x^{2}+20 x+2488\) describes the number, in thousands, of high school graduates in the United States \(x\) years after 1993 a. According to this polynomial, how many students will graduate from U.S. high schools in \(2003 ?\) b. Factor the polynomial. c. Use the factored form of the polynomial in part (b) to find the number of high school graduates in \(2003 .\) Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Explain.

Step-by-Step Solution

Answer

a. There will be 3288 thousand graduates in 2003. b. The factored form is \(4(2x^2 + 5x + 622)\). c. Using the factored form of the polynomial also results in 3288 thousand graduates in 2003 but doesn't inherently prove the correctness of the factorization.

Step 1: Evaluate the polynomial for 2003

In 2003, it will be \(2003 - 1993 = 10\) years after 1993. Substituting \(x = 10\) into the polynomial \(8x^2 + 20x + 2488\) gives \(8(10)^2 + 20(10) + 2488 = 3288\) thousand graduates.

Step 2: Factor the polynomial

Road to factoring always starts with searching for common factors. In this case, we can factor out 4 from each term: \(4(2x^2 + 5x + 622)\). Unfortunately, the resulting quadratic expression \(2x^2 + 5x + 622\) cannot be factored further using integers.

Step 3: Use factored form to find graduates in 2003

Using the factored form of the polynomial, we would substitute \(x = 10\) and should calculate to the same number of graduates for 2003. Doing that, \(4(2(10)^2 + 5(10) + 622) = 3288\) thousand graduates. Indeed, as expected, we get the same number as in step 1.

Step 4: Discuss whether this proves the factorization

We get the same answer for the number of graduates both before and after factoring the polynomial. However, this doesn't inherently prove the factorization is correct but it does shows consistency in the mathematics involved. Factoring is proved correct by showing that it cannot be factored further.

Key Concepts

Factoring

Factoring is a mathematical process used to express a polynomial as a product of its simplest parts. In the world of polynomials, it means breaking down a complicated expression into simpler factors that, when multiplied together, give us the original polynomial. It is like reverse distributing or, putting it another way, finding the pieces that make up the bigger puzzle.

For example, starting with the polynomial \(8x^2 + 20x + 2488\), the first step in factoring is to look for common factors across all terms. In this case, each term can be divided by 4, resulting in \(4(2x^2 + 5x + 622)\).

Sometimes, after factoring out the greatest common factor, what's left is a quadratic expression that can be further factored. However, in our example, \(2x^2 + 5x + 622\) does not break down further using integers. This means our factorization stops here at \(4(2x^2 + 5x + 622)\). It's important to remember that fully factoring a polynomial helps simplify calculations and find roots more easily.

Quadratic Expression

A quadratic expression is a polynomial of degree two. It usually takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The expression involves a squared term, which is what makes it 'quadratic'.

In the given exercise, within the polynomial \(8x^2 + 20x + 2488\), the part \(2x^2 + 5x + 622\) serves as the quadratic expression. This shows a typical structure with \(a = 2\), \(b = 5\), and \(c = 622\).

• The main feature of a quadratic is its parabolic graph, which can open upwards or downwards based on the sign of \(a\).
• Quadratics often require methods such as factoring, completing the square, or using the quadratic formula to solve equations related to them.

Here, even though the quadratic \(2x^2 + 5x + 622\) doesn't factor nicely with integers, it still represents the complex part of the original polynomial that dictates its parabolic nature.

Evaluate Polynomial

Evaluating a polynomial involves substituting a specific value for the variable and simplifying the expression to find a result. This is crucial for understanding how the polynomial behaves at certain points or, as in this exercise, in specific years.

To evaluate the polynomial \(8x^2 + 20x + 2488\) for the year 2003, we determine that 2003 is 10 years after 1993. Consequently, substituting \(x = 10\) yields \(8(10)^2 + 20(10) + 2488 = 3288\). This result tells us that in 2003, there are 3288 thousand high school graduates.

• Evaluating polynomials is a straightforward way to calculate values for real-world scenarios based on the mathematical model provided.
• By evaluating before and after factoring, it also validates the consistency and correctness of the steps taken in simplification.

Number of High School Graduates

The polynomial \(8x^2 + 20x + 2488\) is a mathematical model representing the number of high school graduates, in thousands, years after 1993. Such models help in estimating or predicting real-world data.

This polynomial indicates not just a simple straight-line growth but includes a quadratic term as well, suggesting that the number of graduates changes in a way influenced by both linear and quadratic factors over time.

• By evaluating the polynomial at specific points, like the example year 2003, we can mathematically predict that 3288 thousand students graduate that year.
• Having a reliable model like this helps educators and policymakers plan accordingly for resources and infrastructures needed for such numbers.

Real-world applications of polynomials, such as predicting graduate numbers, highlight the significance of understanding polynomials in both abstract and practical contexts.