Problem 9 - Algebra for College Students

Question

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=7, \quad P(x)=x+7$$

Step-by-Step Solution

Answer

x=7 is not a zero of the function P(x).

Step 1: Understand the given function and value

The function provided is \(P(x) = x + 7\) and we need to check if \(x = 7\) is a zero of this function.

Step 2: Substitute x with the given value in the function

Replace \(x\) with \(7\) in the function: \[P(7) = 7 + 7\]

Step 3: Calculate the result

Perform the calculation: \[P(7) = 7 + 7 = 14\]

Step 4: Determine if the result is zero

Check if the calculated result \(14\) is equal to zero. Since \(14 eq 0\), \(x = 7\) is not a zero of the function \(P(x)\).

Key Concepts

Evaluating Functions

In mathematics, evaluating a function means finding the function's output for a specific input value.
Let's say you have a function denoted as \( f(x) \). To evaluate \( f(x) \) at a specific value like \( x = 3 \), you substitute 3 for \( x \) in the function's equation and simplify.
For example, if \( f(x) = x^2 + 2x \), then \( f(3) = 3^2 + 2(3) = 9 + 6 = 15 \).
This process helps you understand how the function behaves at specific points.

Polynomials

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
Polynomials can have different degrees based on the highest power of the variable. For example:

• The polynomial \( P(x) = x^2 + 3x + 2 \) is of degree 2.
• The polynomial \( P(x) = x + 7 \) is of degree 1.

Polynomials are common in algebra and have various applications in different fields such as physics, economics, and engineering. They can be simple, like \( P(x) = x + 7 \), or more complex.

Zeros of a Function

The zeros of a function are the values of \( x \) that make the function equal to zero.
To find the zeros of a function, you solve the equation \( f(x) = 0 \).
For instance, for the polynomial \( P(x) = x + 7 \), to find the zero, set \( P(x) \) equal to zero:
\[ x + 7 = 0 \]
Solve for \( x \): \[ x = -7 \]
Therefore, the zero of the function \( P(x) = x + 7 \) is \( x = -7 \).
Zeros are essential in graphing and understanding the behavior of functions, especially polynomials.