Problem 44 - Algebra 2

Question

Which of the following is a zero of the function \(f(x)=12 x^{5}-5 x^{3}+2 x-9 ?\) A. \(-6\) B. \(\frac{3}{8}\) C. \(-\frac{2}{3}\) D. 1

Step-by-Step Solution

Answer

Option D, \(x = 1\), is a zero of the function.

Step 1 - Substitute each candidate into the function

The function is given as \(f(x) = 12x^5 - 5x^3 + 2x - 9\). A zero of the function is any value of \(x\) for which \(f(x) = 0\). We will substitute each option (A, B, C, and D) into the function to see which one results in zero.

Step 2 - Evaluate for Option A: \(x = -6\)

Substitute \(-6\) into the function: \[f(-6) = 12(-6)^5 - 5(-6)^3 + 2(-6) - 9\]. Calculate each term of the expression.

Step 3 - Evaluate for Option B: \(x = \frac{3}{8}\)

Substitute \(\frac{3}{8}\) into the function: \[f\left(\frac{3}{8}\right) = 12\left(\frac{3}{8}\right)^5 - 5\left(\frac{3}{8}\right)^3 + 2\left(\frac{3}{8}\right) - 9\]. Calculate each term of the expression.

Step 4 - Evaluate for Option C: \(x = -\frac{2}{3}\)

Substitute \(-\frac{2}{3}\) into the function: \[f\left(-\frac{2}{3}\right) = 12\left(-\frac{2}{3}\right)^5 - 5\left(-\frac{2}{3}\right)^3 + 2\left(-\frac{2}{3}\right) - 9\]. Calculate each term of the expression.

Step 5 - Evaluate for Option D: \(x = 1\)

Substitute \(1\) into the function: \[f(1) = 12(1)^5 - 5(1)^3 + 2(1) - 9\]. Simplifying gives \[12 - 5 + 2 - 9 = 0\]. Since the expression equals zero, \(x = 1\) is a zero of the function.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving variables and coefficients, consisting of terms in the form of a whole number power. It can have constants, linear, quadratic, cubic terms, and beyond.
For example, the polynomial function given in the exercise is \(f(x) = 12x^5 - 5x^3 + 2x - 9\). Each term consists of a coefficient and a power of \(x\):

• The term \(12x^5\) is of degree 5 (cubic term).
• The term \(-5x^3\) is of degree 3.
• The term \(2x\) is of degree 1 (linear term).
• The constant term \(-9\) is of degree 0.

In polynomials, the degree of the term with the highest exponent indicates the degree of the polynomial, which in this case is 5. Understanding polynomials is crucial because they describe a broad class of functions and are used extensively in algebra, calculus, and many applied fields.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to find the result. In simpler terms, you replace the variable(s) in the function with the given value to compute the corresponding outcome.
In this exercise, we evaluated the polynomial function for different values of \(x\) to determine if they are zeros of the function. A zero of a function is a value where \(f(x) = 0\). Each candidate value from the problem's options was substituted:\

• For \(x = -6\), \(x = \frac{3}{8}\), \(x = -\frac{2}{3}\), and \(x = 1\), split the polynomial into terms, calculate each part, and add the results to find \(f(x)\).

Function evaluation is an essential skill that helps in analyzing and graphing functions, solving equations, and modeling real-life scenarios.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific value or another expression. This method can simplify the process of solving equations and finding values like zeros.
In our context, the substitution method was applied to see which value made the polynomial zero:

• Substitute each candidate into \(f(x)\) and simplify to see which results in \(f(x) = 0\).
• When \(x = -6\), test it by computing \(f(-6)\).
• Continue with all candidates until the zero \(x = 1\) is found when \(f(1) = 0\).

This method is particularly useful because it provides a clear, step-by-step process to determine specific outcomes, making it easier to interpret the behavior of functions across different scenarios.