Problem 100 - Algebra A Combined Function

Question

The polynomial \(-30 x^{2}+180 x+210\) represents the approximate number of visitors (in thousands) per year to the White House during \(2003-2007\). In this polynomial, \(x\) represents the years since 2003 . (Source: Based on data from the National Park Service) a. Find the approximate number of visitors to the White House in \(2005 .\) To do so, let \(x=2\) and evaluate \(-30 x^{2}+180 x+210\). b. Find the approximate number of visitors to the White House in 2006 . c. Factor out the GCF from the polynomial \(-30 x^{2}+180 x+210\)

Step-by-Step Solution

Answer

In 2005, 450,000 visitors; in 2006, 480,000 visitors; GCF factor is \(30(-x^2 + 6x + 7)\).

Step 1: Identify the Value of x for 2005

To find the number of visitors in the year 2005, we need to determine the value of \(x\) corresponding to that year. Since \(x\) represents the number of years since 2003, for 2005, \(x = 2005 - 2003 = 2\).

Step 2: Evaluate the Polynomial for 2005

Substitute \(x = 2\) into the polynomial \(-30x^2 + 180x + 210\) and calculate. \[-30(2)^2 + 180(2) + 210 = -30(4) + 360 + 210 = -120 + 360 + 210 = 450\]Thus, the number of visitors in 2005 is approximately 450,000.

Step 3: Identify the Value of x for 2006

Next, identify the value of \(x\) for the year 2006. For 2006, since it's 3 years after 2003, \(x = 2006 - 2003 = 3\).

Step 4: Evaluate the Polynomial for 2006

Substitute \(x = 3\) into the polynomial and calculate:\[-30(3)^2 + 180(3) + 210 = -30(9) + 540 + 210 = -270 + 540 + 210 = 480\]The approximate number of visitors in 2006 is 480,000.

Step 5: Find the Greatest Common Factor (GCF)

To factor out the GCF from \(-30x^2 + 180x + 210\), first identify the GCF of the coefficients -30, 180, and 210. - The GCF of 30, 180, and 210 is 30.- Therefore, factor 30 out of the polynomial:\[-30x^2 + 180x + 210 = 30(-x^2 + 6x + 7)\]Thus, the polynomial factored by the GCF is \(30(-x^2 + 6x + 7)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a way of simplifying a polynomial by finding the largest number that divides each term without a remainder. This is a crucial step in making polynomials more manageable in calculations. For example, let's look at the polynomial \(-30x^2 + 180x + 210\). First, we examine the coefficients:

• -30
• 180
• 210

To find the GCF, we need to determine the highest integer that can go into all these numbers evenly. Start by listing the factors:

• -30: 1, 2, 3, 5, 6, 10, 15, 30
• 180: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30
• 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 30

By comparing these factors, we see that the GCF is 30. Once you identify the GCF, you can simplify the polynomial by factoring it out. In this exercise, factoring 30 gives us \(30(-x^2 + 6x + 7)\). This is a cleaner form and a simpler foundation for further manipulation.

Factoring Polynomials

Factoring polynomials involves breaking them down into the product of simpler polynomials. Typically, this is done by first finding the greatest common factor (GCF) and then using it for further factoring.Let's consider the polynomial from our example: \(-30x^2 + 180x + 210\). After finding the GCF, which is 30, we factored it out to get \(30(-x^2 + 6x + 7)\). This not only simplifies the polynomial but also makes it easier to analyze and use.The polynomial inside the parentheses, \(-x^2 + 6x + 7\), is now simpler and might be easier to factor further using other techniques, such as factoring by grouping or using the quadratic formula when applicable.Factoring helps in reducing expressions to their simplest form, making complex equations more digestible and easier to solve.

Evaluating Polynomials

Evaluating a polynomial means finding its value for a given input. This involves substituting a number for the variable and computing the result.In our exercise, we evaluated the polynomial \(-30x^2 + 180x + 210\) for specific years by substituting values for \(x\). Let's look at how this works step-by-step:

• To find the number of visitors in 2005, we used \(x = 2\) (because 2005 is 2 years after 2003). Substitute and calculate:\[-30(2)^2 + 180(2) + 210 = -120 + 360 + 210 = 450\]This results in approximately 450,000 visitors.
• Similarly, for 2006, we set \(x = 3\) (since it's 3 years after 2003):\[-30(3)^2 + 180(3) + 210 = -270 + 540 + 210 = 480\]This gives about 480,000 visitors.

Evaluating polynomials is essential in analyzing trends, making predictions, and solving real-world problems, as shown in this exercise for determining visitor numbers.