Problem 33 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-3 \text { a factor of } x^{4}-81 ? $$

Step-by-Step Solution

Answer

Yes, \( x-3 \) is a factor of \( x^4 - 81 \).

Step 1: Understand the Factor Theorem

The factor theorem states that a polynomial \( f(x) \) has a factor \( x - c \) if and only if \( f(c) = 0 \). If \( f(c) = 0 \), then \( x - c \) is a factor of \( f(x) \).

Step 2: Identify the Polynomial and the Factor

The polynomial given is \( f(x) = x^4 - 81 \) and the potential factor is \( x - 3 \). According to the Factor Theorem, we need to determine if \( f(3) = 0 \).

Step 3: Substitute \( x = 3 \) into the Polynomial

Substitute \( x = 3 \) into the polynomial equation to find \( f(3) \):\[ f(3) = 3^4 - 81 \]

Step 4: Calculate \( f(3) \)

Now calculate the expression:\[ 3^4 = 81 \]Thus, \[ f(3) = 81 - 81 = 0 \].

Step 5: Conclusion from the Calculation

Since \( f(3) = 0 \), by the Factor Theorem, \( x - 3 \) is indeed a factor of \( x^4 - 81 \).

Key Concepts

Polynomial Factorization

Polynomial factorization is a process where a polynomial expression is written as a product of its factors. Factors of a polynomial are expressions of lower degree, which when multiplied together, give back the original polynomial. This technique is not only limited to simplifying expressions but also helps in solving polynomial equations easily.
To factorize polynomials, one can use various methods such as:

• Factoring by grouping: This involves grouping terms with common factors and simplifying.
• Using special products: Recognizing patterns such as difference of squares can make factoring straightforward.
• Applying the Factor Theorem: This theorem is valuable for determining factors that are linear (such as \(x - c\)).

Understanding how to factor polynomials is essential in both basic algebraic manipulation and in more complex areas like calculus. It simplifies expressions significantly, making them easier to work with.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators (such as addition, subtraction, multiplication, and division). In an algebraic expression, each term can consist of constants and/or variables raised to a power.
For instance, in the expression \(3x^2 + 2x - 5\), we have three distinct terms:

• \(3x^2\) includes a variable raised to a power with a coefficient of 3.
• \(2x\) is a linear term, featuring a variable without an exponent.
• -5 is a constant term, with no variable involved.

Algebraic expressions are unique because they can describe complex relationships and models. An essential skill is simplifying these expressions by combining like terms, factoring, or expanding, depending on the goals of your problem.

Polynomials

Polynomials are a specific category of algebraic expressions that are integral to algebra. They consist of terms that are combined using addition and subtraction, where each term is made up of a coefficient and a variable raised to a non-negative integer exponent. A polynomial can be a single term or a sum of several terms.
The basic structure of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0\), where each \(a_i\) represents a coefficient and \(n\) is a non-negative integer which is the degree of the polynomial.
Polynomials have several important characteristics:

• Degree: The highest power of the variable, which indicates the polynomial's maximum complexity.
• Leading Coefficient: The coefficient of the term with the highest power, playing a key role in the polynomial’s end behavior.
• Zeros/Roots: These are the solutions to the polynomial equation when set equal to zero.

Understanding these properties is crucial for graphing polynomials, solving polynomial equations, and conducting polynomial regression analysis.