Determining Factors Of Polynomial Function F(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14

JU07/10/2025 08, 2025 by THE IDEN 82 views

Determining Factors of a Polynomial Function: A Comprehensive Guide to f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14

In this article, we will delve into the process of determining the factors of a polynomial function. Specifically, we will focus on the function f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14. We will explore how to identify whether certain expressions are factors of this polynomial and provide a step-by-step guide to help you understand the underlying concepts. We will analyze three statements:

I. 3x - 7 is a factor II. x + 2 is a factor III. x + 1 is a factor

and determine which of these statements are true.

Understanding the Factor Theorem

Before we dive into the specifics of our function, it's crucial to understand the Factor Theorem. The Factor Theorem is a fundamental concept in algebra that links the roots of a polynomial to its factors. It states that for a polynomial f(x), if f(a) = 0 for some value a, then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then f(a) = 0. This theorem provides a powerful tool for identifying factors of polynomials.

To effectively utilize the Factor Theorem, we can use synthetic division or direct substitution. Synthetic division is a streamlined method for dividing a polynomial by a linear expression (x - a). If the remainder after synthetic division is zero, then (x - a) is a factor. Direct substitution involves plugging the value a into the polynomial f(x). If the result is zero, then (x - a) is a factor. Both methods serve the same purpose, and the choice between them often comes down to personal preference or the specific problem at hand.

In the context of our problem, we will use the Factor Theorem to test whether the given expressions (3x - 7, x + 2, and x + 1) are factors of f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14. This involves finding the roots corresponding to each expression and checking if the function evaluates to zero at those roots. By applying this theorem systematically, we can accurately determine the factors of the polynomial.

Testing Statement I: Is 3x - 7 a Factor?

To determine if 3x - 7 is a factor of f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14, we need to find the root of the expression 3x - 7. We set 3x - 7 = 0 and solve for x:

3x - 7 = 0 3x = 7 x = 7/3

Now, we substitute x = 7/3 into the function f(x):

f(7/3) = 3(7/3)^4 - 13(7/3)^3 + 11(7/3)^2 + 13(7/3) - 14

Calculating this value:

f(7/3) = 3(2401/81) - 13(343/27) + 11(49/9) + 13(7/3) - 14 f(7/3) = 2401/27 - 4459/27 + 539/9 + 91/3 - 14 f(7/3) = (2401 - 4459 + 1617 + 819 - 378)/27 f(7/3) = 0/27 = 0

Since f(7/3) = 0, according to the Factor Theorem, 3x - 7 is indeed a factor of f(x). This confirms that Statement I is true.

This meticulous calculation demonstrates the application of the Factor Theorem. By finding the root of the potential factor and substituting it into the polynomial, we can definitively determine whether the expression is a factor. The fact that f(7/3) = 0 provides concrete evidence that 3x - 7 divides f(x) evenly, leaving no remainder. This process is crucial for factoring higher-degree polynomials and understanding their behavior.

Testing Statement II: Is x + 2 a Factor?

To determine if x + 2 is a factor of f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14, we need to find the root of the expression x + 2. We set x + 2 = 0 and solve for x:

x + 2 = 0 x = -2

Now, we substitute x = -2 into the function f(x):

f(-2) = 3(-2)^4 - 13(-2)^3 + 11(-2)^2 + 13(-2) - 14

Calculating this value:

f(-2) = 3(16) - 13(-8) + 11(4) + 13(-2) - 14 f(-2) = 48 + 104 + 44 - 26 - 14 f(-2) = 156

Since f(-2) = 156, which is not equal to 0, x + 2 is not a factor of f(x). This indicates that Statement II is false.

This result further illustrates the importance of the Factor Theorem. The non-zero result of f(-2) definitively shows that x + 2 does not divide f(x) evenly. This highlights the precision of the theorem in determining factors. The calculation demonstrates that a potential factor must result in a zero value when substituted into the polynomial; otherwise, it cannot be considered a factor.

Testing Statement III: Is x + 1 a Factor?

To determine if x + 1 is a factor of f(x) = 3x^4 - 13x^3 + 11x^2 + 13x - 14, we find the root of the expression x + 1. We set x + 1 = 0 and solve for x:

x + 1 = 0 x = -1

Now, we substitute x = -1 into the function f(x):

f(-1) = 3(-1)^4 - 13(-1)^3 + 11(-1)^2 + 13(-1) - 14

Calculating this value:

f(-1) = 3(1) - 13(-1) + 11(1) + 13(-1) - 14 f(-1) = 3 + 13 + 11 - 13 - 14 f(-1) = 0

Since f(-1) = 0, according to the Factor Theorem, x + 1 is a factor of f(x). Thus, Statement III is true.

This result reinforces the power of the Factor Theorem as a tool for identifying factors. The zero value of f(-1) provides clear evidence that x + 1 is a factor of f(x). This process highlights how a simple substitution can yield significant information about the polynomial's structure. The consistent application of the Factor Theorem allows us to methodically analyze potential factors and build a complete understanding of the polynomial's factorization.

Conclusion: Identifying the Correct Statements

Based on our analysis:

Statement I: 3x - 7 is a factor (True)
Statement II: x + 2 is a factor (False)
Statement III: x + 1 is a factor (True)

Therefore, the correct answer is A. I and III.

In summary, we have demonstrated how to use the Factor Theorem to determine whether given expressions are factors of a polynomial function. By finding the roots of the expressions and substituting them into the function, we can efficiently identify factors. This method is a cornerstone of polynomial algebra and provides a reliable way to analyze and factor complex expressions.

Understanding and applying the Factor Theorem is essential for solving polynomial equations, simplifying algebraic expressions, and gaining a deeper understanding of the behavior of polynomial functions. This comprehensive guide has provided a clear and step-by-step approach to determining factors, empowering you to tackle similar problems with confidence. The methodical application of the Factor Theorem, as demonstrated in this article, is a fundamental skill in algebra and is crucial for further studies in mathematics. The ability to accurately identify factors is not only important for academic success but also has practical applications in various fields that rely on mathematical modeling and analysis.