Problem 25 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-1 \text { a factor of } 4 x^{3}-13 x^{2}+21 x-12 \text { ? } $$

Step-by-Step Solution

Step 1: Identify the Polynomial and Potential Factor

The given polynomial is \( 4x^3 - 13x^2 + 21x - 12 \). The potential factor in question is \( x-1 \). We need to verify whether \( x-1 \) is a factor using the Factor Theorem.

Step 2: Understand the Factor Theorem

The Factor Theorem states that \( x-c \) is a factor of a polynomial \( f(x) \) if and only if \( f(c) = 0 \). In this case, we need to verify if \( f(1) = 0 \) for the polynomial \( f(x) = 4x^3 - 13x^2 + 21x - 12 \).

Step 3: Evaluate the Polynomial at \( x = 1 \)

Substitute \( x = 1 \) into the polynomial:\[ f(1) = 4(1)^3 - 13(1)^2 + 21(1) - 12 \]}{

Key Concepts

Polynomial Factorization

Polynomial factorization is a process of breaking down a complex polynomial into simpler polynomials whose product is equal to the original. This is an essential concept in algebra because it allows for easier manipulation and understanding of polynomials.
In this exercise, we are investigating if the polynomial \( 4x^3 - 13x^2 + 21x - 12 \) can be factored by \( x-1 \). To accomplish this, we use the factor theorem.Polynomial factorization is like finding the ingredients that make up a cake, where the products (ingredients) are used to create a whole (cake). Simplifying polynomials through factorization is helpful for solving equations, graphing polynomial functions, and analyzing mathematical strengths. With effective factorization, the polynomial can be expressed in a form that reveals its roots and other properties.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial function at a given point. In simpler terms, it is like plugging numbers into a formula to see what result you get. It plays a crucial role in checking the validity of factors using the factor theorem.For our exercise, we need to evaluate the polynomial \( f(x) = 4x^3 - 13x^2 + 21x - 12 \) at \( x = 1 \). This involves substituting \( x \) with 1:

• Compute \( 4(1)^3 = 4 \)
• Compute \( -13(1)^2 = -13 \)
• Compute \( +21(1) = 21 \)
• Subtract 12: \( -12 \)

We calculate:\[ f(1) = 4 - 13 + 21 - 12 \]This results in \( f(1) = 0 \), confirming that \( x = 1 \) is indeed a root, making \( x-1 \) a factor of the polynomial.

Factorization Steps

Factorization steps are structured procedures to determine whether a term like \( x-1 \) is a factor of a polynomial. Let's break down the steps used in solving this specific problem.Start by identifying the polynomial, as was given: \( 4x^3 - 13x^2 + 21x - 12 \). Recognize the potential factor: \( x-1 \). Then, employ the factor theorem:

• Substitute \( x = 1 \) into the polynomial to evaluate: \( f(x) = 4x^3 - 13x^2 + 21x - 12 \)
• Calculate \( f(1) \)

After performing these calculations, if you find that \( f(1) = 0 \), it means \( x-1 \) is a factor. For this exercise, this calculated value was 0, confirming the factorization.
This step-by-step method is vital for systematically confirming factors within polynomials, aiding in clearer understanding and solving of algebraic problems. Following these structured steps ensures no detail is overlooked in the validation process.