Finding Zeros Of Polynomials P(x) = 5x³ - X² - 15x + 3 In Exact Form

In mathematics, finding the zeros of a polynomial function is a fundamental problem with applications spanning various fields, including engineering, physics, and computer science. The zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. These zeros provide crucial information about the behavior and characteristics of the polynomial function. In this article, we will delve into the process of finding the zeros of the polynomial function p(x) = 5x³ - x² - 15x + 3 in exact form. We will explore different techniques and strategies to identify these zeros, which are the points where the graph of the polynomial intersects the x-axis. Understanding how to find zeros is essential for analyzing polynomial functions and solving related problems.

The zeros of a polynomial p(x) are the values of x for which p(x) = 0. These values represent the x-intercepts of the polynomial's graph. Finding the zeros of a polynomial is equivalent to solving the equation p(x) = 0. The zeros can be real or complex numbers, and they provide valuable information about the polynomial's behavior and properties. For instance, the number of real zeros corresponds to the number of times the graph of the polynomial crosses the x-axis. Additionally, the zeros can help determine the intervals where the polynomial is positive or negative. Understanding the concept of polynomial zeros is crucial for various mathematical applications, including curve sketching, equation solving, and mathematical modeling. Polynomial zeros are also closely related to the factors of the polynomial, as each zero corresponds to a linear factor. By finding the zeros, we can factor the polynomial, which can simplify further analysis and calculations. In summary, understanding polynomial zeros is a fundamental concept in algebra and calculus, with broad applications in mathematics and related fields.

Techniques for Finding Zeros

Several techniques can be employed to find the zeros of a polynomial, depending on its degree and coefficients. For linear and quadratic polynomials, we have direct methods like solving linear equations or using the quadratic formula. However, for higher-degree polynomials, the process becomes more complex. One common approach is factoring, where we try to express the polynomial as a product of simpler factors. If we can factor the polynomial into linear factors, we can easily find the zeros by setting each factor equal to zero and solving for x. Another technique is the Rational Root Theorem, which provides a list of possible rational roots based on the coefficients of the polynomial. This theorem can be particularly helpful for polynomials with integer coefficients. We can also use numerical methods, such as the Newton-Raphson method, to approximate the zeros of a polynomial. These methods involve iterative calculations to converge towards the zeros. In some cases, we may need to use a combination of these techniques to find all the zeros of a polynomial. For instance, we might use the Rational Root Theorem to find some rational roots and then use polynomial division to reduce the degree of the polynomial. This can simplify the process of finding the remaining zeros. In summary, finding the zeros of a polynomial often requires a combination of algebraic techniques, numerical methods, and strategic problem-solving skills.

The Rational Root Theorem is a powerful tool for finding potential rational zeros of a polynomial with integer coefficients. The theorem states that if a polynomial p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ has a rational zero p/q (where p and q are integers with no common factors other than 1), then p must be a factor of the constant term a₀, and q must be a factor of the leading coefficient aₙ. In other words, the possible rational zeros are of the form ±(factors of a₀) / (factors of aₙ). For our polynomial p(x) = 5x³ - x² - 15x + 3, the constant term is 3, and the leading coefficient is 5. The factors of 3 are ±1 and ±3, and the factors of 5 are ±1 and ±5. Therefore, the possible rational zeros are ±1, ±3, ±1/5, and ±3/5. By testing these values, we can determine which ones are actual zeros of the polynomial. This can be done through direct substitution or using synthetic division. The Rational Root Theorem significantly narrows down the possibilities for rational zeros, making it a valuable technique for solving polynomial equations. It is especially useful when combined with other methods, such as polynomial division, to find all the zeros of the polynomial. In summary, the Rational Root Theorem provides a systematic way to identify potential rational zeros, which can then be verified through testing.

Factoring by grouping is a technique used to factor polynomials with four or more terms. It involves grouping terms together in pairs and then factoring out the greatest common factor (GCF) from each pair. If the resulting expressions have a common factor, we can factor it out, leading to a factored form of the polynomial. This technique is particularly useful when there is no obvious common factor for all terms in the polynomial. For the given polynomial p(x) = 5x³ - x² - 15x + 3, we can group the first two terms and the last two terms: (5x³ - x²) + (-15x + 3). From the first group, we can factor out x², giving us x²(5x - 1). From the second group, we can factor out -3, giving us -3(5x - 1). Now, we have x²(5x - 1) - 3(5x - 1). Notice that (5x - 1) is a common factor. Factoring it out, we get (5x - 1)(x² - 3). This factored form simplifies the process of finding the zeros of the polynomial. Factoring by grouping is a powerful technique that can be applied to various polynomials, making it an essential tool in algebra. It often requires careful observation and strategic grouping of terms to identify common factors. In summary, factoring by grouping is a valuable method for factoring polynomials, especially those with four or more terms, by identifying and extracting common factors from grouped terms.

Now that we have factored the polynomial p(x) = 5x³ - x² - 15x + 3 into (5x - 1)(x² - 3), we can find the zeros by setting each factor equal to zero and solving for x. First, let's consider the factor (5x - 1). Setting this equal to zero gives us 5x - 1 = 0. Solving for x, we add 1 to both sides to get 5x = 1, and then divide by 5 to get x = 1/5. This is one of the zeros of the polynomial. Next, let's consider the factor (x² - 3). Setting this equal to zero gives us x² - 3 = 0. Solving for x, we add 3 to both sides to get x² = 3. Taking the square root of both sides gives us x = ±√3. Thus, we have two more zeros: x = √3 and x = -√3. These zeros are irrational, as they involve the square root of a non-perfect square. In summary, the zeros of the polynomial p(x) = 5x³ - x² - 15x + 3 are 1/5, √3, and -√3. These values represent the points where the graph of the polynomial intersects the x-axis. Finding these zeros allows us to fully understand the behavior and characteristics of the polynomial function.

The zeros of the polynomial p(x) = 5x³ - x² - 15x + 3 in exact form are 1/5, √3, and -√3. Exact form means that we express the zeros using radicals and fractions, rather than decimal approximations. This is important because decimal approximations can introduce rounding errors and may not be as precise as the exact values. The zero 1/5 is a rational number, and it is already in its simplest form. The zeros √3 and -√3 are irrational numbers, as they cannot be expressed as a ratio of two integers. They are expressed using the square root symbol, which represents the exact value. To verify these zeros, we can substitute them back into the original polynomial and check if the result is zero. For example, p(1/5) = 5(1/5)³ - (1/5)² - 15(1/5) + 3 = 0, which confirms that 1/5 is a zero. Similarly, p(√3) = 5(√3)³ - (√3)² - 15(√3) + 3 = 0, and p(-√3) = 5(-√3)³ - (-√3)² - 15(-√3) + 3 = 0, which confirm that √3 and -√3 are also zeros. Expressing the zeros in exact form is crucial for maintaining accuracy and precision in mathematical calculations. It also allows for a deeper understanding of the nature of the zeros, whether they are rational or irrational. In summary, the exact zeros of the polynomial are 1/5, √3, and -√3, providing a complete and accurate solution to the problem.

In this article, we have successfully found the zeros of the polynomial function p(x) = 5x³ - x² - 15x + 3 in exact form. We began by understanding the concept of polynomial zeros and their significance in analyzing polynomial functions. We then explored various techniques for finding zeros, including factoring by grouping and the Rational Root Theorem. By applying these techniques, we factored the polynomial into (5x - 1)(x² - 3) and identified the zeros as 1/5, √3, and -√3. These zeros represent the x-intercepts of the polynomial's graph and provide valuable information about its behavior. We emphasized the importance of expressing the zeros in exact form to maintain accuracy and precision. The process of finding zeros involves a combination of algebraic techniques, strategic problem-solving, and a deep understanding of polynomial functions. This skill is essential for various mathematical applications, including curve sketching, equation solving, and mathematical modeling. In conclusion, finding the zeros of a polynomial is a fundamental problem in mathematics with broad applications, and the techniques discussed in this article provide a solid foundation for solving such problems. The zeros of p(x) = 5x³ - x² - 15x + 3 are therefore 1/5, √3, and -√3.

The zeros of p(x): 1/5, √3, -√3