Problem 8 - Algebra for College Students

Question

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=3, \quad P(x)=x-3$$

Step-by-Step Solution

Answer

Yes, x = 3 is a zero of the function P(x) = x - 3.

Step 1: Substitute the value of x into the function

Substitute the given value of {x}: {3} into the function {P(x) = x - 3}. This means we need to replace {x} with {3} in the function {P(x)}.

Step 2: Evaluate the function

Now, evaluate the function at {x = 3}: {P(3) = 3 - 3}.

Step 3: Check whether the result is zero

Compute the expression: {3 - 3 = 0}p(3)=0 p(3)(Therefore, {x=3} is a zero of the function {P(x) = x - 3}. .)

Key Concepts

Function Evaluation

Function evaluation is an important concept in mathematics. It helps us understand a function's behavior at different points. To evaluate a function, you simply substitute a given value for each variable in the function.
Let’s take a closer look at the given exercise and solution. The function provided is \( P(x) = x - 3 \) and we are asked to evaluate it at \( x = 3 \).
Substitute \( x = 3 \) into the function:
\( P(3) = 3 - 3 \)
Now, simplify the expression:
\( P(3) = 0 \)
Since the result is zero, it means \( x = 3 \) is a zero of the function.

This evaluation helps us identify specific points where the function equals zero, indicating the zeros of the function.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients, involving terms in the form of \( ax^n \), where \( n \) is a non-negative integer. In general, they can be written as:
\( P(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \)
Here, each term consists of a variable raised to a power with a coefficient. Considering our given function \( P(x) = x - 3 \), it is a first-degree polynomial, also known as a linear function.

Key properties of polynomial functions include:

• They are continuous and smooth.
• The degree of the polynomial determines its shape and behavior.
• Zeros of the polynomial are points where the function value equals zero.

Understanding these properties helps in analyzing and solving polynomial equations efficiently.

Solving Equations

Solving equations is a fundamental aspect of algebra. It involves finding the value of variables that satisfy the equation. For polynomial equations, we aim to determine the zeros or roots of the function.

In our exercise, solving the equation \( P(x) = x - 3 \) means finding the value of \( x \) that makes \( P(x) = 0 \). Here are the steps to solve such equations:

• Write down the function: \( P(x) = x - 3 \).
• Set the function equal to zero: \( x - 3 = 0 \).
• Solve for \( x \): Add 3 to both sides to get \( x = 3 \).

Thus, \( x = 3 \) is a zero of the function. Understanding how to solve equations equips students with the skills needed to handle more complex algebraic expressions and real-world problems.