Problem 8 - Precalculus : Building Concepts and Connections

Question

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$f(x)=x^{2}+9, x=-3,3$$

Step-by-Step Solution

Answer

Neither \(x=-3\) nor \(x=3\) are zeros of the given polynomial \(f(x)=x^{2}+9\).

Step 1: Substitute \(x=-3\) into the polynomial

Firstly, substitute \(x=-3\) into the polynomial \(f(x)=x^{2}+9\). This results in \(f(-3)=(-3)^{2}+9 = 9 + 9 = 18\). So, \(x=-3\) is not a zero of the polynomial.

Step 2: Substitute \(x=3\) into the polynomial

Secondly, substitute \(x=3\) into the polynomial \(f(x)=x^{2}+9\). This results in \(f(3)=(3)^{2}+9 = 9 + 9 = 18\). So, \(x=3\) is also not a zero of the polynomial.

Key Concepts

Zeros of Polynomials

Polynomials are expressions in mathematics that can have constants, variables, and exponents. Finding the zeros of a polynomial involves determining the values for which the polynomial evaluates to zero. These zeros are often the points where the graph of the polynomial intersects the x-axis. Knowing the zeros helps us understand the root structure of the polynomial and its behavior.

• A zero of a polynomial can be found by solving the equation \(f(x) = 0\).
• Zeros are not restricted to integer or real values; they can be complex numbers too.
• Graphically, zeros correspond to x-intercepts of the polynomial function.

In the original exercise, values of \(x\) were substituted into the polynomial \(f(x)=x^{2}+9\) to check if they zeroed the function, which they did not in either case.

Evaluating Polynomials

Evaluating a polynomial means calculating the value of the polynomial function for a given value of \(x\). It is a straightforward process of substitution and then simplifying. This is a critical step in determining if a particular \(x\)-value is a zero for the polynomial.

• Start by substituting the value of \(x\) into the polynomial.
• Next, carry out any arithmetic operations such as addition, subtraction, multiplication, or exponents.
• Finally, determine the result to see if it equals zero for finding zeros.

In the exercise provided, we evaluated the polynomial for \(x = -3\) and \(x = 3\) by performing these steps, leading to an outcome of 18 in both cases, which means neither value was a zero.

Substitution Method

Substitution is a technique used to simplify the process of evaluating expressions or equations by replacing variables with their numeral values. This is particularly handy in complex polynomials where it helps streamline calculations and identify the solution.

• Begin with substituting the value into the polynomial, replacing the variable \(x\).
• Simplify the expression step-by-step to avoid errors.
• Verify your steps to ensure accuracy in results.

In our exercise, we utilized substitution to check if the given numbers \(x = -3\) and \(x = 3\) made the polynomial \(f(x) = x^{2} + 9\) zero, proving not to be zeros after evaluation.