Problem 13 - Precalculus: Functions and Graphs

Question

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). $$f(x)=x^{3}+x^{2}-2 x+12 ; \quad c=-3$$

Step-by-Step Solution

Answer

Yes, \(x + 3\) is a factor of \(f(x)\) since \(f(-3) = 0\).

Step 1: Identify the equation and factor

We are given the polynomial function \(f(x) = x^3 + x^2 - 2x + 12\) and need to show that \(x - c\), where \(c = -3\), is a factor of \(f(x)\). This means that \(f(c)\) should equal zero by the factor theorem.

Step 2: Substitute c into the polynomial

Substitute \(c = -3\) into the function: \[ f(-3) = (-3)^3 + (-3)^2 - 2(-3) + 12 \]

Step 3: Evaluate the substituted expression

Calculate each term after substitution:- \((-3)^3 = -27\)- \((-3)^2 = 9\)- \(-2(-3) = 6\)- Adding the constant \(12\)Thus, \[ f(-3) = -27 + 9 + 6 + 12 \]

Step 4: Compute the final value

Add the results of the previous calculations:\[ -27 + 9 + 6 + 12 = 0 \]

Step 5: Determine if \(x + 3\) is a factor

Since \(f(-3) = 0\), the factor theorem confirms that \(x + 3\) is a factor of \(f(x)\).

Key Concepts

Polynomial Functions

Polynomial functions form the backbone of many mathematical studies. A polynomial is an expression made up of terms where each term is a product of a constant and a variable raised to a non-negative integer power. For example, the polynomial function given in the original exercise is

• This specific function is a cubic polynomial because the highest degree of its variable \(x\) is 3.
• Each term is composed of a constant (also known as a coefficient) multiplied by the variable raised to a power. In this case: \( x^3, x^2, -2x, \) and the constant term of 12.

The degree of the polynomial is the highest power of \(x\). The polynomial

• Serves as a template to create a range of outputs by inputting different values for \(x\).
• As mentioned, polynomials can be classified into linear, quadratic, cubic, and so forth, based on their highest exponent, and each type has unique characteristics and graph shapes.

A well-understood and widely used property of polynomial functions is their continuity and differentiability across their domain, which contributes to broader applications in calculus and analysis.

Remainder Theorem

The Remainder Theorem is an essential perspective when analyzing polynomials. It simplifies the process of evaluating polynomial expressions.

• When we divide a polynomial \(f(x)\) by a linear divisor \((x - c)\), the remainder of this division is precisely \(f(c)\).

This observation leads directly into the Factor Theorem, a specific case where

• If \(f(c) = 0\), then \(x - c\) is a factor of \(f(x)\).

So if a specific \(c\) makes \(f(c)\) equal to zero, then the polynomial \(f(x)\) is perfectly divisible by \(x - c\). By confirming that \(f(-3) = 0\), in the original solution, we have utilized the Remainder Theorem to affirm the Factor Theorem's assertion that \(x + 3\) is a factor.

Synthetic Substitution

Synthetic substitution is a compact technique used to evaluate polynomials at a given value. It is especially useful when using the Factor Theorem or Remainder Theorem. In general, synthetic substitution streamlines the process by focusing directly on the coefficients of the polynomial.

• Identify the coefficients of each term. For the polynomial \(x^3 + x^2 - 2x + 12\), these coefficients are \(1, 1, -2,\) and \(12\).
• Carry out a method that computes a series of operations exclusively on these coefficients in relation to the constant \(c\), which in this case is -3. This involves multiplying the constant by a number and adding it to the next coefficient repeatedly.

This method aids in confirming the result effectively without computing each individual term manually, making it a quick verification approach to determine if \(f(c) = 0\). Although synthetic substitution isn't explicitly shown in the steps of the original solution, it's a complementary method that demonstrates why \(x + 3\) is a factor, showcasing its utility in polynomial evaluation.