Remainder Theorem Determine Zeros Of Polynomial G(x) = X³ - 3x² - 10x + 30
In this article, we will explore how to use the Remainder Theorem to determine if a given number, denoted as c, is a zero of a polynomial function. The Remainder Theorem is a fundamental concept in algebra that provides a powerful method for evaluating polynomials and identifying their roots. Specifically, we will consider the polynomial function g(x) = x³ - 3x² - 10x + 30 and investigate whether c = -3 and c = -√10 are zeros of this polynomial. Understanding how to apply the Remainder Theorem is crucial for polynomial factorization, root finding, and solving various algebraic problems. By the end of this article, you will have a solid grasp of this theorem and its applications.
The Remainder Theorem states that if a polynomial g(x) is divided by x - c, then the remainder is g(c). In simpler terms, to find the remainder when a polynomial g(x) is divided by a linear divisor x - c, you can simply evaluate g(c). This theorem provides a quick and efficient way to find the remainder without performing long division. Moreover, it connects the remainder of polynomial division with the value of the polynomial at a specific point. This connection is incredibly useful for determining if a number is a zero of the polynomial. If g(c) = 0, then c is a zero of the polynomial, meaning that (x - c) is a factor of g(x). The Remainder Theorem is not just a shortcut for finding remainders; it is a bridge that links polynomial division, evaluation, and factorization.
Let's apply the Remainder Theorem to determine if c = -3 is a zero of the polynomial g(x) = x³ - 3x² - 10x + 30. According to the Remainder Theorem, we need to evaluate g(-3). This means we substitute -3 for x in the polynomial expression and simplify. So, we have:
g(-3) = (-3)³ - 3(-3)² - 10(-3) + 30
First, we calculate the powers:
(-3)³ = -27 (-3)² = 9
Next, we substitute these values back into the expression:
g(-3) = -27 - 3(9) - 10(-3) + 30
Now, we perform the multiplications:
g(-3) = -27 - 27 + 30 + 30
Finally, we add and subtract the numbers:
g(-3) = -54 + 60 g(-3) = 6
Since g(-3) = 6, which is not equal to zero, we can conclude that c = -3 is not a zero of the polynomial g(x) = x³ - 3x² - 10x + 30. This result tells us that (x + 3) is not a factor of g(x). The Remainder Theorem provides a straightforward method to verify this without having to perform polynomial long division or synthetic division.
Now, let's use the Remainder Theorem to check if c = -√10 is a zero of the polynomial g(x) = x³ - 3x² - 10x + 30. We need to evaluate g(-√10). This involves substituting -√10 for x in the polynomial expression:
g(-√10) = (-√10)³ - 3(-√10)² - 10(-√10) + 30
First, we calculate the powers:
(-√10)³ = (-√10)² * (-√10) = 10 * (-√10) = -10√10 (-√10)² = 10
Next, we substitute these values back into the expression:
g(-√10) = -10√10 - 3(10) - 10(-√10) + 30
Now, we perform the multiplications:
g(-√10) = -10√10 - 30 + 10√10 + 30
Finally, we simplify the expression by combining like terms:
g(-√10) = (-10√10 + 10√10) + (-30 + 30) g(-√10) = 0 + 0 g(-√10) = 0
Since g(-√10) = 0, we can conclude that c = -√10 is a zero of the polynomial g(x) = x³ - 3x² - 10x + 30. This means that (x + √10) is a factor of g(x). The Remainder Theorem efficiently confirms this, illustrating its usefulness in identifying zeros of polynomials, especially those involving radicals.
The Remainder Theorem has significant implications in polynomial algebra. One of the most important is its connection to the Factor Theorem, which is a direct corollary. The Factor Theorem states that a polynomial g(x) has a factor (x - c) if and only if g(c) = 0. This theorem is invaluable for factoring polynomials. When we find a zero c of a polynomial, we immediately know a factor (x - c), which simplifies the process of breaking down the polynomial into simpler components. Furthermore, the Remainder Theorem provides a method for synthetic division, a streamlined process for dividing a polynomial by a linear factor. Synthetic division is particularly useful for finding the quotient when dividing by (x - c) and, by extension, for reducing the degree of the polynomial once a factor is identified.
The applications of the Remainder Theorem extend to polynomial factorization and root finding. Factoring a polynomial involves expressing it as a product of simpler polynomials. The Remainder Theorem helps in this process by allowing us to identify linear factors. Once a factor is found, the polynomial can be divided by that factor to reduce its degree, making it easier to find additional factors. Root finding, which involves determining the values of x for which g(x) = 0, is another critical application. Each zero of a polynomial corresponds to a root of the polynomial equation g(x) = 0. The Remainder Theorem provides a direct method to test potential roots. By evaluating the polynomial at a specific value, we can quickly determine if that value is a root. These applications highlight the Remainder Theorem's role in simplifying complex algebraic problems and providing a systematic approach to polynomial analysis.
In conclusion, the Remainder Theorem is a powerful tool for determining if a given number c is a zero of a polynomial. By evaluating the polynomial g(x) at x = c, we can quickly ascertain whether (x - c) is a factor of g(x). This method is both efficient and insightful, offering a direct link between polynomial evaluation and factorization. In the cases we examined, we found that c = -3 is not a zero of g(x) = x³ - 3x² - 10x + 30, while c = -√10 is a zero. Understanding and applying the Remainder Theorem is essential for mastering polynomial algebra and solving related problems effectively. This theorem not only simplifies the process of finding remainders but also serves as a cornerstone for polynomial factorization and root finding.
