Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$g(x)=\frac{x^{2}-8 x+12}{x^{2}+4}$$

Step-by-Step Solution

Answer

The zeros of the function \(g(x) = \frac{x^{2}-8x+12}{x^{2}+4}\) are x = 2 and x = 6

Step 1: Solve the Quadratic Equation

Solve the quadratic equation by making the numerator equal to zero and finding the values of x. This will gives the zeros of function \(g(x)\):So, we have \(x^{2} - 8x + 12 = 0\)

Step 2: Factorize the Quadratic Equation

Factor the equation in the form \((x-a)(x-b) = 0\). Here, a and b are two numbers that multiply to give 12 (the third term) and add up to give -8 (the coefficient of x). If we check the pair of factors of 12, we can see (2,6) fits these criteria. Thus, the equation can be factorized as:\((x - 2)(x - 6) = 0\)

Step 3: Find the Values of x

To find the zeros, set each factor equal to zero and solve for 'x':\((x - 2) = 0 -> x = 2\)\((x - 6) = 0 -> x = 6\)

Step 4: Verify With a Graphing Utility

The zeros calculated above can be verified by plotting the function \(g(x) = \frac{x^{2}-8 x+12}{x^{2}+4}\) graphically. The zeros of the function, or the x-intercepts, should be on x=2 and x=6.

Key Concepts

Zeros of Functions

Discovering zeros of a function helps us analyze its behavior and graphical representation. Zeros are the values of \( x \) for which the function equals zero. For a rational function, these occur when the numerator is zero but not the denominator. This is because the denominator being zero would make the function undefined.
To find the zeros of a function like \( g(x)=\frac{x^{2}-8x+12}{x^{2}+4} \), we set the numerator \( x^{2} - 8x + 12 \) equal to zero. Solving this equation yields the points where the graph intersects the x-axis. Thus, zeros are an essential aspect of comprehending the function's graph.

Quadratic Equations

Quadratic equations play a fundamental role in algebra and involve expressions of the form \( ax^2 + bx + c = 0 \). Solutions to quadratic equations can often be found by factoring, using the quadratic formula, or completing the square.
In our exercise, the quadratic equation is \( x^{2} - 8x + 12 = 0 \). We aim to solve this equation to find the zeros of the rational function. Solving such equations usually results in two values for \( x \), indicating the points where the function is zero.
Understanding how to manipulate and solve quadratic equations is crucial for graphing and analyzing functions in various contexts.

Graphing Utilities

Graphing utilities are valuable tools for visualizing functions and verifying algebraic solutions. They help plot equations to check the location of zeros, intercepts, asymptotes, and other features quickly.
To verify our solution for \( g(x) = \frac{x^{2}-8x+12}{x^{2}+4} \), plotting the graph helps ensure that the calculated zeros \( x = 2 \) and \( x = 6 \) coincide with the x-intercepts. Many graphing utilities include options to trace the graph, allowing users to see function values at particular points.
Using graphing technology in mathematical problem-solving assists in making an abstract concept more concrete and understandable.

Factorization

Factorization breaks down an expression into a product of its simplest factors. For quadratic expressions like \( x^{2} - 8x + 12 \), factorization involves expressing them as \( (x-a)(x-b) \), where a and b are derived from specific relationships involving the coefficients.
In our case, finding integers whose product is 12 (constant term) and whose sum is -8 (coefficient of x), leads to the factors \( (x-2)\) and \( (x-6) \). Solving \( (x-2)(x-6) = 0 \) directly provides the zeros of the function.
Mastering factorization simplifies solving equations and helps reveal the structure of the expressions involved, aiding in deeper mathematical insights.