Problem 24 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-5 \text { a factor of } 8 x^{2}-47 x+32 ? $$

Step-by-Step Solution

Answer

No, \( x-5 \) is not a factor because \( f(5) \neq 0 \).

Step 1: Understanding the Factor Theorem

The factor theorem states that a polynomial \( f(x) \) has a factor \( x-c \) if and only if \( f(c) = 0 \). This means to determine if \( x-5 \) is a factor, we need to check if substituting 5 into the polynomial results in zero.

Step 2: Substituting to Find \( f(5) \)

We have the polynomial \( f(x) = 8x^2 - 47x + 32 \). Substitute \( x = 5 \) into this polynomial: \[f(5) = 8(5)^2 - 47(5) + 32.\]

Step 3: Calculating \( f(5) \)

Calculate \( f(5) = 8(25) - 47(5) + 32 \). First, find \( 8(25) = 200 \) and \( 47(5) = 235 \). Now substitute these values back into the equation: \[ f(5) = 200 - 235 + 32 \].

Step 4: Simplifying to See If It Equals Zero

Simplify the expression: \[ f(5) = 200 - 235 + 32 = -35 + 32 = -3 \]. Since \( f(5) = -3 \) and not zero, \( x-5 \) is not a factor of the polynomial.

Key Concepts

Polynomial Factorization

Polynomial factorization is a key process in algebra that involves expressing a polynomial as the product of its factors. Think of it like breaking down a large number into its smaller prime components or parts that multiply to make the original number. This process allows us to simplify polynomials, solve polynomial equations more easily, and see insights into their structure.

To factor polynomials, we often use methods such as:

• Finding common factors, where you look for terms that can be factored out of the polynomial.
• Using special formulas, like differences of squares or perfect square trinomials.
• Applying the factor theorem, which tells us that if a polynomial evaluated at a certain value equals zero, then that value is a root, and the related factor is part of the polynomial's factorization.

These methods help break down complex expressions and solve equations in a straightforward way. Understanding how to factor is foundational to mastering algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) that form an equation or relationship. When dealing with algebraic expressions, it’s important to simplify them, just like cleaning up and arranging your room for better usability.

In the context of the Factor Theorem, an algebraic expression like a polynomial can be tested for factors by substituting specific values to see if they yield zero, which helps determine if the expression can be factored further.

Algebraic expressions can be manipulated in various ways:

• Combining like terms by adding or subtracting coefficients of the same variable power.
• Distributive property, where you distribute a multiplied value across terms inside parentheses.
• Substitution, which is entering specific numbers for variables to evaluate the expression.

This foundational understanding helps in solving equations and modeling real-world scenarios.

Roots of Polynomials

Roots of polynomials are the values of the variable that make the polynomial equal zero. In simpler terms, they are the solutions to the polynomial equation. Finding these roots is a fundamental task in algebra as it gives insights into the behavior and solutions of polynomial functions.

For any given polynomial, its roots can be found using the factor theorem. If substituting a value into the polynomial makes it zero, then that value is a root, and the corresponding factor is \(x - \ ext{value}\). This process not only confirms roots but also helps in factorizing the polynomial completely.

• For example, in our exercise, checking if \(x-5\) is a factor involves calculating whether substituting 5 results in zero.
• Roots can be real or complex numbers depending on the polynomial’s coefficients.
• Each root corresponds to at least one factor of the polynomial.

Understanding roots is crucial for graphing polynomial functions and solving algebraic equations effectively.