For what value of [tex]$x$[/tex] would the quotient [tex]$\left(2 x^2+3 x-4\right) \div(x+4)$[/tex] not make sense?
A. 4
B. -2
C. -4
D. 0
Answer (1)
A rational expression is undefined when its denominator equals zero.
Set the denominator x + 4 to zero: x + 4 = 0 .
Solve for x : x = − 4 .
The quotient is undefined when x = − 4 .
Explanation
Understanding the Problem The given expression is a quotient of two polynomials: x + 4 2 x 2 + 3 x − 4 . A rational expression (a fraction with polynomials) is undefined when the denominator is equal to zero. Our goal is to find the value of x that makes the denominator, x + 4 , equal to zero.
Setting the Denominator to Zero To find the value of x that makes the denominator zero, we set the denominator equal to zero and solve for x : x + 4 = 0
Solving for x Subtracting 4 from both sides of the equation, we get: x = − 4
Finding the Undefined Value Therefore, the quotient x + 4 2 x 2 + 3 x − 4 is undefined when x = − 4 .
Examples
Understanding when a rational expression is undefined is crucial in various real-world applications. For instance, in electrical engineering, the impedance of a circuit can be represented as a rational function. Determining the values at which the impedance becomes undefined helps engineers identify resonant frequencies, which are critical for designing filters and amplifiers. Similarly, in physics, rational functions can model the behavior of waves, and identifying undefined points helps predict phenomena like wave interference or resonance. In economics, rational functions can model cost-benefit ratios, and understanding when these ratios become undefined can help in making informed decisions about investments or resource allocation.
