Problem 14 - Precalculus: Functions and Graphs

Question

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). $$f(x)=x^{3}+x^{2}-11 x+10 ; \quad c=2$$

Step-by-Step Solution

Answer

Yes, \( x-2 \) is a factor of \( f(x) \) since \( f(2) = 0 \).

Step 1: Substitute the value of c into the function

Use the given value of \( c = 2 \) and substitute 2 into the function \( f(x) = x^3 + x^2 - 11x + 10 \). This means we need to calculate \( f(2) \).

Step 2: Calculate f(2)

Substitute 2 for \( x \) in the equation:\[ f(2) = (2)^3 + (2)^2 - 11(2) + 10 \].Calculate step-by-step: - \( (2)^3 = 8 \).- \( (2)^2 = 4 \).- \( 11(2) = 22 \).Combine these results to get \( 8 + 4 - 22 + 10 \).

Step 3: Simplify the equation

Add and subtract the values: \( 8 + 4 = 12 \), then \( 12 - 22 = -10 \), and finally \( -10 + 10 = 0 \).

Step 4: Conclude using the Factor Theorem

The factor theorem states that \( x-c \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \). Since \( f(2) = 0 \), it confirms that \( x-2 \) is a factor of \( f(x) \).

Key Concepts

Polynomial Functions

In mathematics, polynomial functions are expressions involving a variable raised to various powers, with coefficients linked to each term. Consider \[ f(x) = x^3 + x^2 - 11x + 10 \]Here, the polynomial is of degree 3, because the highest power of the variable \( x \) is 3. Each term in a polynomial consists of a coefficient (like the numbers in front of \( x \)) and a variable raised to an exponent. These terms can be combined by addition or subtraction.
Polynomial functions can describe a wide range of real-world problems, from physics to economics. They are essential in algebraic operations due to their versatile properties. Moreover, each polynomial function is defined over a range of possible inputs, called the domain, typically all real numbers for most basic functions.

Division of Polynomials

The division of polynomials is a process that resembles the long division we learn in arithmetic. Given a polynomial \( f(x) \) and a divisor, the goal is to express \( f(x) \) as the product of this divisor and another polynomial, with possibly a remainder. In the exercise, we used the factor theorem to help with this division.

• If \( x-c \) is a factor, the division will leave no remainder.
• If \( x-c \) is not a factor, there will be some leftover (the remainder).

This was evidenced when we calculated \( f(2) \) and it equaled 0, showing \( x-2 \) divides the polynomial without remainder, confirming it as a factor.
Understanding polynomial division helps solve algebraic equations and understand more complex mathematical challenges, particularly when simplifying rational expressions or solving equations.

Roots of Equations

Finding the roots of a polynomial equation involves determining the values of the variable that make the polynomial equal zero. These values are crucial because they are the solutions to the equation.For the function \( f(x) = x^3 + x^2 - 11x + 10 \), using the factor theorem, we identified that \( x = 2 \) is a root. This means when we substitute \( x = 2 \) into the polynomial, it equals zero. It's like finding the x-intercepts of a polynomial graph, where the curve crosses the x-axis.

• Identifying roots helps simplify complex polynomial functions.
• The roots provide deep insights into the behavior of the polynomial's graph.

Finding roots can sometimes be challenging, requiring techniques like factoring, using the quadratic formula, or synthetic division. For real-life problems, knowing a polynomial's roots can assist in designing systems or predicting outcomes that rely on algebraic relationships.