Factoring Polynomials Expressing G(x) = X³ - 6x² + X + 34 As Linear Factors And Finding Zeros

\begin{array}{l}
g(x)=x^3-6 x^2+x+34 \\
g(x)=\square
\end{array}

List all the zeros of the function. (Enter your answers as a comma-separated list. Enter all answers using the simplest form.)

Factoring Polynomials into Linear Factors and Finding Zeros

In this article, we will explore the process of expressing a polynomial as a product of linear factors and subsequently identifying the zeros of the function. This is a fundamental concept in algebra with wide-ranging applications in various fields of mathematics and beyond. Specifically, we will focus on the polynomial g(x) = x³ - 6x² + x + 34, demonstrating how to factor it into linear factors and determine its zeros. Understanding these techniques is crucial for solving polynomial equations, analyzing the behavior of polynomial functions, and tackling more complex mathematical problems.

Understanding Polynomial Factorization

Factoring a polynomial involves breaking it down into simpler expressions (factors) that, when multiplied together, yield the original polynomial. Linear factors are polynomials of degree one, typically in the form (x - a), where 'a' is a constant. The zeros of a polynomial function are the values of x for which the function equals zero. These zeros correspond to the x-intercepts of the polynomial's graph. To find the zeros, we set the polynomial equal to zero and solve for x. Factoring a polynomial into linear factors makes it easier to identify these zeros.

Polynomial factorization is a crucial skill in algebra, allowing us to simplify complex expressions and solve equations. The linear factors of a polynomial provide direct insight into its roots or zeros, which are the values of x that make the polynomial equal to zero. When dealing with a cubic polynomial like g(x) = x³ - 6x² + x + 34, finding these factors can seem challenging at first. However, by systematically applying techniques like the Rational Root Theorem and synthetic division, we can break down the polynomial into its linear components. This process not only reveals the zeros of the function but also helps us understand the polynomial's behavior and graph. The ability to express a polynomial as a product of linear factors is a powerful tool for solving equations, analyzing functions, and tackling various mathematical problems, making it a fundamental concept in algebra and beyond.

Techniques for Factoring Polynomials

Several techniques can be employed to factor polynomials, including:

1. Rational Root Theorem: This theorem helps identify potential rational roots (zeros) of the polynomial. It states that if a polynomial has integer coefficients, then any rational root 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.
2. Synthetic Division: This is a streamlined method for dividing a polynomial by a linear factor (x - a). If the remainder is zero, then (x - a) is a factor of the polynomial, and 'a' is a zero of the function.
3. Factoring by Grouping: This technique involves grouping terms in the polynomial and factoring out common factors. It is particularly useful for polynomials with four or more terms.
4. Quadratic Formula: If a polynomial can be reduced to a quadratic form (ax² + bx + c), the quadratic formula can be used to find its roots.

Rational Root Theorem is the first step in finding rational zeros of the polynomial, helping to narrow down the possibilities for trial and error. This theorem states that any rational zero of a polynomial must be a factor of the constant term divided by a factor of the leading coefficient. For our polynomial, g(x) = x³ - 6x² + x + 34, the constant term is 34 and the leading coefficient is 1. This means any rational roots must be factors of 34, which are ±1, ±2, ±17, and ±34. This drastically reduces the number of potential zeros we need to test.

Synthetic division then allows us to efficiently test these potential roots. This method provides a quick way to divide a polynomial by a linear factor (x - a) and determine if 'a' is a zero. If the remainder after synthetic division is zero, then (x - a) is indeed a factor of the polynomial, and 'a' is a zero. This process helps us systematically identify the linear factors of the polynomial. Factoring by grouping can be effective when dealing with polynomials with four or more terms, but it's not directly applicable to our cubic polynomial until we've identified one linear factor. Lastly, the quadratic formula comes into play once we've reduced the cubic polynomial to a quadratic equation. By applying the Rational Root Theorem, synthetic division, and potentially the quadratic formula, we can methodically factor the polynomial and find all its zeros.

Applying the Techniques to g(x) = x³ - 6x² + x + 34

Let's apply these techniques to our polynomial g(x) = x³ - 6x² + x + 34.

1. Rational Root Theorem: The possible rational roots are the factors of 34 divided by the factors of 1, which are ±1, ±2, ±17, and ±34.
2. Synthetic Division: We can test these potential roots using synthetic division. Let's start with x = -2:

-2 | 1 -6 1 34
| -2 16 -34
----------------
1 -8 17 0

Since the remainder is 0, x = -2 is a root, and (x + 2) is a factor.

3. Factoring the Quadratic: The result of the synthetic division gives us the quadratic x² - 8x + 17. This quadratic does not factor easily, so we use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a x = (8 ± √((-8)² - 4 * 1 * 17)) / 2 x = (8 ± √(64 - 68)) / 2 x = (8 ± √(-4)) / 2 x = (8 ± 2i) / 2 x = 4 ± i

Applying the Rational Root Theorem to g(x) = x³ - 6x² + x + 34, we identify a set of potential rational roots: ±1, ±2, ±17, and ±34. These values represent the possible candidates for zeros of the polynomial, based on the factors of the constant term (34) and the leading coefficient (1). This initial step is crucial because it provides a manageable list of values to test, rather than an infinite range of possibilities. The Rational Root Theorem effectively narrows our search, making the subsequent steps more efficient. Without this theorem, finding the zeros of the polynomial would be a much more challenging task.

Using synthetic division to test these potential roots, we find that x = -2 is indeed a root of the polynomial. The synthetic division process not only confirms this but also provides us with the quotient, which is a quadratic polynomial. This is a significant step forward because it reduces the cubic polynomial to a more manageable quadratic form. The coefficients of the quotient are derived from the synthetic division process, and in this case, they represent the coefficients of the quadratic x² - 8x + 17. This quadratic represents the remaining factor of the original cubic polynomial, after we've extracted the linear factor (x + 2). The efficiency of synthetic division in both testing potential roots and reducing the polynomial's degree makes it a vital technique in polynomial factorization.

The resulting quadratic x² - 8x + 17 does not factor easily using traditional methods. This is where the quadratic formula comes into play. The quadratic formula provides a direct way to find the roots of any quadratic equation, regardless of whether it can be factored using simpler techniques. Applying the quadratic formula to x² - 8x + 17 reveals that the roots are complex numbers: 4 + i and 4 - i. This indicates that the original cubic polynomial has one real root (x = -2) and two complex roots. The presence of complex roots is not uncommon in polynomial equations, and the quadratic formula ensures that we can find these roots accurately. This comprehensive approach, combining the Rational Root Theorem, synthetic division, and the quadratic formula, allows us to completely factor the polynomial and identify all its zeros, both real and complex.

Expressing g(x) as a Product of Linear Factors

Therefore, we can express g(x) as a product of linear factors:

g(x) = (x + 2)(x - (4 + i))(x - (4 - i))

Listing the Zeros of the Function

The zeros of the function are: -2, 4 + i, 4 - i.

Expressing g(x) as a product of linear factors is the culmination of our efforts. By identifying the zeros of the polynomial, we can rewrite it in a factored form that clearly shows its linear components. Each zero corresponds to a linear factor: the real root -2 gives us the factor (x + 2), while the complex roots 4 + i and 4 - i give us the factors (x - (4 + i)) and (x - (4 - i)), respectively. Multiplying these linear factors together reconstructs the original polynomial. This factored form is not just a mathematical representation; it also provides valuable insights into the polynomial's behavior. For example, it directly reveals the x-intercepts of the polynomial's graph (in the case of real roots) and helps in understanding its overall shape and characteristics. The ability to express a polynomial in its factored form is a key step in solving equations, analyzing functions, and applying polynomial concepts in various mathematical and scientific contexts.

Listing the zeros of the function provides a complete picture of the polynomial's roots. In this case, g(x) has three zeros: one real zero at x = -2 and two complex zeros at x = 4 + i and x = 4 - i. These zeros are the values of x for which the polynomial equals zero, and they play a crucial role in understanding the polynomial's behavior. The real zero corresponds to the point where the polynomial's graph intersects the x-axis, while the complex zeros do not have a direct graphical representation on the real plane. However, all zeros, both real and complex, are essential for fully characterizing the polynomial. Knowing the zeros allows us to solve polynomial equations, factor the polynomial, and analyze its properties. The zeros are fundamental to understanding the polynomial's structure and its relationship to other mathematical concepts.

Conclusion

In this article, we successfully factored the polynomial g(x) = x³ - 6x² + x + 34 into linear factors and identified its zeros. We employed the Rational Root Theorem, synthetic division, and the quadratic formula to achieve this. The final factored form is g(x) = (x + 2)(x - (4 + i))(x - (4 - i)), and the zeros are -2, 4 + i, and 4 - i. This process demonstrates the power of these algebraic techniques in analyzing and solving polynomial equations.

Factoring polynomials and finding their zeros are essential skills in algebra and have wide-ranging applications in mathematics, science, and engineering. The techniques we've explored in this article provide a systematic approach to tackling polynomial equations, allowing us to break down complex expressions into simpler components. The ability to factor polynomials and identify their zeros is crucial for solving equations, analyzing functions, and modeling real-world phenomena. From determining the trajectory of a projectile to designing electrical circuits, the concepts and methods discussed here are fundamental tools for problem-solving in various fields. This article serves as a comprehensive guide to mastering these techniques, empowering students and professionals alike to confidently tackle polynomial problems and apply them in diverse contexts.