Solving And Factoring Polynomials Finding Zeros Of F(x) = 5x^3 - X^2 - 65x + 13

Jul 11, 2025 by Admin 80 views

Solving Polynomial Equations and Factoring Let f(x) = 5x^3 - x^2 - 65x + 13

This article delves into the process of finding the zeros of a cubic polynomial function and factoring it into linear factors. We will use the Rational Root Theorem, synthetic division, and the quadratic formula to achieve this. Our specific example is the function f(x) = 5x^3 - x^2 - 65x + 13.

(a) Finding Rational and Other Zeros of f(x) = 5x^3 - x^2 - 65x + 13

To solve f(x) = 0, we first need to identify the potential rational zeros. The Rational Root Theorem is a crucial tool here. This theorem states that if a polynomial has integer coefficients, then any rational zero must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case, the constant term is 13 and the leading coefficient is 5.

Applying the Rational Root Theorem

The factors of the constant term, 13, are ±1 and ±13. The factors of the leading coefficient, 5, are ±1 and ±5. Therefore, the possible rational zeros are:

±1/1 = ±1
±13/1 = ±13
±1/5
±13/5

This gives us a list of potential rational roots: -13, -1, -13/5, -1/5, 1/5, 13/5, 1, 13. To determine which of these are actual zeros, we can use synthetic division or direct substitution. Synthetic division is a more efficient method for testing multiple potential roots.

Using Synthetic Division to Test Potential Zeros

Let's start by testing x = 1/5. Synthetic division involves setting up a table and performing a series of multiplications and additions.

1/5 | 5  -1  -65  13
    |     1   0  -13
    ----------------
      5   0  -65   0

The remainder is 0, which means x = 1/5 is a zero of f(x). The quotient is 5x^2 + 0x - 65, which simplifies to 5x^2 - 65. This quotient represents the reduced polynomial after dividing f(x) by (x - 1/5).

Solving the Reduced Quadratic Equation

Now we have a quadratic equation, 5x^2 - 65 = 0. We can solve this by factoring or using the quadratic formula. Let's factor out the common factor of 5:

5(x^2 - 13) = 0

This implies x^2 - 13 = 0, so x^2 = 13. Taking the square root of both sides, we get x = ±√13. These are the remaining two zeros of the polynomial.

Summarizing the Zeros

Therefore, the zeros of f(x) = 5x^3 - x^2 - 65x + 13 are:

x = 1/5
x = √13
x = -√13

We have found one rational zero (1/5) and two irrational zeros (√13 and -√13).

(b) Factoring f(x) into Linear Factors

Now that we have the zeros, we can factor f(x) into linear factors. Each zero corresponds to a linear factor of the form (x - zero). Since the zeros are 1/5, √13, and -√13, the linear factors are (x - 1/5), (x - √13), and (x + √13).

Constructing the Factored Form

To write the factored form, we multiply these linear factors together, along with the leading coefficient of the original polynomial, which is 5:

f(x) = 5(x - 1/5)(x - √13)(x + √13)

We can further simplify this by distributing the 5 into the first factor:

f(x) = (5x - 1)(x - √13)(x + √13)

Verifying the Factored Form

To verify this, we can expand the factored form and check if it matches the original polynomial:

(5x - 1)(x - √13)(x + √13) = (5x - 1)(x^2 - 13)

Expanding further:

5x^3 - 65x - x^2 + 13 = 5x^3 - x^2 - 65x + 13

This matches the original polynomial, so our factored form is correct.

Final Factored Form

Thus, the factored form of f(x) = 5x^3 - x^2 - 65x + 13 into linear factors is:

f(x) = (5x - 1)(x - √13)(x + √13)

Conclusion

In this article, we have successfully found the rational and irrational zeros of the cubic polynomial function f(x) = 5x^3 - x^2 - 65x + 13. We used the Rational Root Theorem to identify potential rational zeros, synthetic division to test these zeros, and the quadratic formula to find the remaining zeros. Finally, we expressed the polynomial in its factored form as a product of linear factors. This process demonstrates a comprehensive approach to solving polynomial equations and factoring polynomials, skills that are fundamental in algebra and calculus. Understanding these methods allows for deeper analysis and manipulation of polynomial functions in various mathematical and scientific contexts. The ability to find zeros and factor polynomials is essential for solving equations, graphing functions, and modeling real-world phenomena. The Rational Root Theorem serves as a starting point, narrowing down the potential rational solutions, while synthetic division provides an efficient way to test these possibilities. Once the polynomial is reduced to a quadratic, familiar methods such as factoring or the quadratic formula can be applied. The culmination of this process is the expression of the polynomial as a product of linear factors, providing valuable insights into the polynomial's behavior and solutions.

By working through this example, readers can gain a solid understanding of the techniques involved in solving polynomial equations and factoring polynomials. This knowledge empowers them to tackle more complex problems and apply these skills in various mathematical and scientific applications. The importance of each step, from identifying potential rational roots to constructing the factored form, is highlighted in the comprehensive approach presented in this article. The combination of theoretical concepts and practical application ensures a thorough understanding of the subject matter, making it accessible to students and enthusiasts alike. The process not only aids in solving equations but also enhances the ability to analyze and interpret polynomial functions, which are crucial in various fields, including engineering, physics, and economics. The factored form of a polynomial reveals its zeros, which are essential for understanding its behavior and graphical representation. Furthermore, the factored form is often necessary for simplifying algebraic expressions and solving more complex equations involving rational functions. Therefore, mastering the techniques discussed in this article provides a solid foundation for advanced mathematical studies and practical problem-solving.

In summary, this article provides a step-by-step guide to finding the zeros of a cubic polynomial and expressing it in its factored form. The use of the Rational Root Theorem, synthetic division, and quadratic formula are clearly demonstrated, making the process accessible and understandable. The detailed explanations and examples equip readers with the necessary skills to solve similar problems and deepen their understanding of polynomial functions. The relevance of these techniques in various mathematical and scientific contexts is emphasized, highlighting their importance in advanced studies and practical applications. The ability to factor polynomials and find their zeros is a fundamental skill in mathematics, and this article provides a comprehensive resource for mastering this skill.