Finding Zeros Of The Function H(x) = 2x³ - 23x² + 58x + 35
Finding the zeros of a function is a fundamental problem in mathematics with applications across various fields, including engineering, physics, and computer science. In this comprehensive guide, we will delve into the process of finding all the zeros of the cubic function h(x) = 2x³ - 23x² + 58x + 35. We will explore the techniques involved in solving polynomial equations, including the Rational Root Theorem, synthetic division, and the quadratic formula. By understanding these methods, you'll be well-equipped to tackle similar problems and gain a deeper appreciation for the beauty and power of algebra.
Understanding Zeros of a Function
Before we dive into the solution, let's clarify what we mean by the zeros of a function. The zeros of a function, also known as roots or x-intercepts, are the values of x for which the function's output, h(x), is equal to zero. In other words, they are the points where the graph of the function intersects the x-axis. Finding the zeros of a function is equivalent to solving the equation h(x) = 0. For polynomial functions, such as the one we are considering, the zeros can be real numbers or complex numbers.
For the given function, h(x) = 2x³ - 23x² + 58x + 35, we are looking for the values of x that satisfy the equation:
2x³ - 23x² + 58x + 35 = 0
This is a cubic equation, which means it has a degree of 3. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n complex roots, counting multiplicities. Therefore, our cubic equation will have three roots, which may be real or complex, and some roots may be repeated.
The Rational Root Theorem: A Powerful Tool
To find the zeros of our cubic function, we'll start by employing the Rational Root Theorem. This theorem provides a systematic way to identify potential rational roots of a polynomial equation. A rational root is a root that can be expressed as a fraction p/q, where p and q are integers, and q is not zero.
The Rational Root Theorem states that if a polynomial equation with integer coefficients, such as our equation 2x³ - 23x² + 58x + 35 = 0, has rational roots, then they 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 35, and its factors are ±1, ±5, ±7, and ±35.
- The leading coefficient is 2, and its factors are ±1 and ±2.
Therefore, the possible rational roots of our equation are:
±1/1, ±5/1, ±7/1, ±35/1, ±1/2, ±5/2, ±7/2, ±35/2
Simplifying these fractions, we get the following list of potential rational roots:
±1, ±5, ±7, ±35, ±1/2, ±5/2, ±7/2, ±35/2
This list gives us a set of candidate values for the zeros of our function. To determine which of these values are actual roots, we can use synthetic division or direct substitution.
Synthetic Division: A Streamlined Approach
Synthetic division is a highly efficient method for dividing a polynomial by a linear factor of the form (x - c), where c is a constant. It's a streamlined alternative to long division, particularly useful for testing potential roots of a polynomial equation.
To perform synthetic division, we set up a table with the coefficients of the polynomial and the potential root we want to test. Let's start by testing the potential root x = -1/2:
-1/2 | 2 -23 58 35
| -1 12. -35
------------------
2 -24 70 0
The numbers in the bottom row are the coefficients of the quotient polynomial and the remainder. The last number in the bottom row is the remainder. If the remainder is 0, then the potential root is an actual root of the polynomial.
In this case, the remainder is 0, which means that x = -1/2 is a root of the equation 2x³ - 23x² + 58x + 35 = 0. This also means that (x + 1/2) is a factor of the polynomial. To eliminate the fraction, we can say that (2x + 1) is a factor of the polynomial.
The quotient polynomial, obtained from the bottom row of the synthetic division, is 2x² - 24x + 70. This is a quadratic polynomial, and we can find its roots using the quadratic formula or by factoring.
Factoring and the Quadratic Formula
Now that we've found one root, x = -1/2, we can factor the original cubic polynomial as follows:
2x³ - 23x² + 58x + 35 = (2x + 1)(x² - 12x + 35)
To find the remaining roots, we need to solve the quadratic equation x² - 12x + 35 = 0. We can try to factor this quadratic, or we can use the quadratic formula.
The quadratic formula is a general solution for quadratic equations of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 1, b = -12, and c = 35. Plugging these values into the quadratic formula, we get:
x = (12 ± √((-12)² - 4 * 1 * 35)) / (2 * 1) x = (12 ± √(144 - 140)) / 2 x = (12 ± √4) / 2 x = (12 ± 2) / 2
This gives us two solutions:
x = (12 + 2) / 2 = 14 / 2 = 7 x = (12 - 2) / 2 = 10 / 2 = 5
Therefore, the roots of the quadratic equation x² - 12x + 35 = 0 are x = 5 and x = 7.
The Zeros of h(x)
We have now found all three zeros of the function h(x) = 2x³ - 23x² + 58x + 35:
- x = -1/2
- x = 5
- x = 7
These are the values of x for which h(x) = 0. We can write the zeros as a set: {-1/2, 5, 7}.
In conclusion, by applying the Rational Root Theorem, synthetic division, and the quadratic formula, we successfully found all the zeros of the cubic function h(x) = 2x³ - 23x² + 58x + 35. This process demonstrates the power of algebraic techniques in solving polynomial equations and provides a valuable framework for tackling similar problems in the future. Understanding these methods is crucial for anyone working with polynomials and seeking to gain a deeper understanding of mathematical functions.
Importance of Finding Zeros of a Function
Finding the zeros of a function is a crucial aspect of mathematical analysis and has numerous applications in various fields. The zeros, also known as roots or x-intercepts, provide valuable information about the behavior of a function and its relationship with the x-axis. Understanding how to find these zeros is essential for solving equations, analyzing graphs, and modeling real-world phenomena.
One of the primary reasons for finding zeros is to solve equations. When we set a function equal to zero, we are essentially asking:
