Problem 65 - Intermediate Algebra

Question

Use the factor theorem and determine whether the first expression is a factor of \(P(x) .\) See Example 5. $$ x-3 ; P(x)=x^{3}-3 x^{2}+5 x-15 $$

Step-by-Step Solution

Answer

Yes, \(x - 3\) is a factor of \(P(x)\) because \(P(3) = 0\).

Step 1: Understand the Factor Theorem

The Factor Theorem states that \(x - a\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). This means we need to evaluate \(P(3)\) to see if it equals zero.

Step 2: Substitute and Evaluate P(x)

Substitute \(x = 3\) in \(P(x) = x^3 - 3x^2 + 5x - 15\). Calculate \(P(3) = 3^3 - 3(3)^2 + 5(3) - 15\).

Step 3: Perform the Calculations

Compute each term: \(3^3 = 27\), \(3 \times 3^2 = 27\), \(5 \times 3 = 15\). Thus, \(P(3) = 27 - 27 + 15 - 15\).

Step 4: Simplify the Expression

Simplify the expression \(27 - 27 + 15 - 15\) which results in \(0\).

Step 5: Conclusion Based on Factor Theorem

Since \(P(3) = 0\), according to the Factor Theorem, \(x - 3\) is a factor of \(P(x)\).

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions that consist of variables raised to whole number exponents and coefficients. They can take various forms depending on the number and degree of terms they have. In this exercise, we are dealing with a cubic polynomial, which means the highest degree of the variable, in this case, is 3. Hence, the polynomial function can be written as:

• Cubic Polynomial: This is a polynomial of degree 3. For example, the given polynomial is \( P(x) = x^3 - 3x^2 + 5x - 15 \).
• Terms of a Polynomial: The terms in a polynomial are the individual components separated by either addition or subtraction. In our polynomial, those terms are \(x^3\), \(-3x^2\), \(5x\), and \(-15\).

Understanding how polynomials are structured helps in simplifying expressions and in understanding more complex algebraic operations.
Polynomials are essential in algebra, as they form the building blocks for a wide range of calculations, including solving equations and modeling real-world scenarios.

Basics of Algebra and Factor Theorem

Algebra helps us in understanding and solving equations symbolically. The Factor Theorem is a critical concept in algebra that connects factors of polynomials to their roots. Let's break this down further:

• Expressions and Equations: Algebra is about finding the unknown values. Expressions like \(x - 3\) are parts of larger equations used to find when a polynomial equals zero.
• Factor Theorem Description: According to the Factor Theorem, \((x - a)\) is a factor of a polynomial \(P(x)\) if and only if substituting \(x = a\) in \(P(x)\) results in zero.

To implement this theorem, we substitute the value from our factor, here \(a = 3\), into \(P(x)\) and check if the outcome is zero. Algebra allows us to simplify these expressions and find the solution efficiently.

Identifying Roots of Equations

The roots or solutions of polynomials are points where the polynomial equals zero. These are also referred to as the 'zeros' of the polynomial. Finding these roots is an essential step in solving polynomial equations:

• Roots and Zeros: In the given exercise, by confirming \(P(3) = 0\), we've identified \(x = 3\) as a root of the polynomial.
• Finding Roots Using Factor Theorem: The Factor Theorem allows us to check if \((x - 3)\) is a factor by substituting 3 and confirming it results in zero. Thus, \(x = 3\) is a root.

Recognizing roots helps in simplifying equations and solving for unknowns. It’s a vital part of algebra that allows us to analyze and understand the behavior of equations better.
This understanding helps to connect the mathematical theory to practical problem solving.