Problem 40 - Precalculus: Functions and Graphs

Question

Find all values of \(k\) such that \(f(x)\) is divisible by the given linear polynomial. $$f(x)=k^{2} x^{3}-4 k x+3 ; \quad x-1$$

Step-by-Step Solution

Answer

The values of \( k \) are 1 and 3.

Step 1: Understand the Problem Statement

We have a polynomial function \( f(x) = k^2 x^3 - 4kx + 3 \) and we need to find values of \( k \) such that \( f(x) \) is divisible by \( x-1 \). This means that \( x - 1 \) is a factor of \( f(x) \). By the factor theorem, this happens when \( f(1) = 0 \).

Step 2: Use the Factor Theorem

According to the factor theorem, \( x-1 \) will be a factor of \( f(x) \) if and only if \( f(1) = 0 \). Let's substitute \( x = 1 \) in \( f(x) \) and set it equal to zero.

Step 3: Substitute and Simplify

Substitute \( x = 1 \) into \( f(x) \): \[ f(1) = k^2(1)^3 - 4k(1) + 3 = k^2 - 4k + 3 \] Now, set \( f(1) = 0 \): \[ k^2 - 4k + 3 = 0 \]

Step 4: Solve the Quadratic Equation

The equation \( k^2 - 4k + 3 = 0 \) is a quadratic equation in standard form. To solve it, use the quadratic formula: \[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = -4 \), and \( c = 3 \). Calculate: \[ b^2 - 4ac = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4 \] Thus, the solutions are: \[ k = \frac{-(-4) \pm \sqrt{4}}{2 \times 1} = \frac{4 \pm 2}{2} \]

Step 5: Calculate Specific Values of k

Using the calculations from Step 4, we find the values of \( k \): \[ k = \frac{4 + 2}{2} = 3 \] \[ k = \frac{4 - 2}{2} = 1 \] Therefore, \( k = 3 \) or \( k = 1 \).

Key Concepts

Factor Theorem

The Factor Theorem is a powerful tool in algebra that helps us determine whether a given polynomial is divisible by a linear polynomial. In essence, it states that a polynomial \( f(x) \) has \( x - c \) as a factor if and only if substituting \( c \) into the polynomial results in zero, i.e., \( f(c) = 0 \). This theorem is particularly useful because it offers a straightforward method for checking the divisibility of polynomials without having to perform polynomial long division or synthetic division.

• For example, consider \( f(x) = k^2 x^3 - 4kx + 3 \) and we want to know if \( x - 1 \) is a factor.
• According to the Factor Theorem, we plug \( x = 1 \) into \( f(x) \) and check if it equals zero.
• If \( f(1) = 0 \), then \( x - 1 \) is indeed a factor.

Using the factor theorem simplifies solving problems like the one in the exercise, as it negates the need for less efficient methods.

Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. Solving a quadratic equation involves finding the values of \( x \) that make the polynomial equal to zero. These values are known as the roots of the equation. There are various methods for solving quadratic equations, including:

• Factoring
• Completing the square
• The quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In the exercise, once we applied the Factor Theorem and found \( f(1) = k^2 - 4k + 3 \), it resulted in a quadratic equation \( k^2 - 4k + 3 = 0 \). We used the quadratic formula to solve for \( k \), which led us to the values \( k = 3 \) and \( k = 1 \). The quadratic equation is fundamental in algebra and has applications in numerous fields, making it essential for students to understand how to solve these efficiently.

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of variables with constant coefficients. They play a critical role in algebra, providing the foundation for various algebraic theories and practical applications, from physics to finance. A general form of a polynomial function is \( f(x) = a_n x^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \), where \( a_n, a_{n-1}, \, \ldots \, , a_1, a_0 \) are constants and \( a_n eq 0 \). Key features of polynomial functions include:

• Degree: This refers to the highest power of the variable \( x \).
• Coefficients: These are the constants that multiply the powers of \( x \).
• Roots: The solutions or zeroes of the polynomial equation \( f(x) = 0 \).

Understanding polynomials is crucial, as they are often used to model real-world scenarios, including analyzing trends and forecasting future events. In the context of the exercise, the polynomial \( f(x) = k^2 x^3 - 4kx + 3 \) represents a cubic function that varies depending on the value of \( k \). By determining specific values of \( k \), we reveal conditions under which the polynomial behaves in certain ways.

Divisibility in Algebra

Divisibility in algebra involves determining whether one polynomial can be divided by another without a remainder, akin to checking if a number is divisible by another number. In the context of polynomials, understanding divisibility is crucial for simplifying expressions and solving equations effectively. When given a polynomial \( f(x) \), we aim to check if it is divisible by another polynomial like \( x - c \). This exercise often involves using the Factor Theorem, which provides a simple method of checking whether the divisibility condition \( f(c) = 0 \) holds true.

• In our example, we wanted to see if \( f(x) = k^2 x^3 - 4kx + 3 \) was divisible by \( x - 1 \).
• Applying the Factor Theorem allowed us to determine the condition for divisibility by simply evaluating \( f(1) \) and setting it to zero.
• Solving the resulting quadratic equation gave us the specific values of \( k \) for which the divisibility holds: \( k = 3 \) and \( k = 1 \).

Understanding algebraic divisibility enables students to simplify polynomial equations and expressions more easily and makes solving complex algebraic problems more manageable.