Finding X-Intercepts Of F(x) = -2x⁵ - X⁴ + 10x³ + 5x² A Step-by-Step Guide
To determine the x-intercepts of a function, we need to find the values of x for which f(x) = 0. In this detailed exploration, we will focus on the function f(x) = -2x⁵ - x⁴ + 10x³ + 5x². We will systematically factor the polynomial, identify the roots, and confirm these roots graphically. This comprehensive guide aims to provide a clear understanding of how to find all x-intercepts of polynomial functions through factoring.
1. Factoring Out Common Terms
The first step in factoring any polynomial is to look for common factors among all the terms. In our function, f(x) = -2x⁵ - x⁴ + 10x³ + 5x², we can observe that each term contains at least x². Factoring out x² simplifies the polynomial and makes it easier to work with:
f(x) = x²(-2x³ - x² + 10x + 5)
This initial factoring step is crucial as it reduces the degree of the polynomial inside the parentheses, making subsequent factoring steps more manageable. Factoring out the greatest common factor not only simplifies the expression but also immediately reveals one or more roots of the polynomial. In this case, we see that x² is a factor, which tells us that x = 0 is a root with a multiplicity of 2. This means the graph of the function will touch the x-axis at x = 0 but will not cross it.
2. Factoring by Grouping
After factoring out the common term, we are left with a cubic polynomial inside the parentheses: -2x³ - x² + 10x + 5. Factoring cubic polynomials can sometimes be challenging, but a common technique is factoring by grouping. This method involves grouping terms in pairs and factoring out common factors from each pair. Let's apply this to our cubic polynomial:
- Group the terms: (-2x³ - x²) + (10x + 5)
- Factor out common factors from each group:
- From the first group, factor out -x²: -x²(2x + 1)
- From the second group, factor out 5: 5(2x + 1)
- Notice that both groups now have a common factor of (2x + 1). Factor this out: (2x + 1)(-x² + 5)
Factoring by grouping is a powerful technique that simplifies the process of finding roots for higher-degree polynomials. The key to successful grouping is to arrange the terms in such a way that common factors can be easily identified and factored out. This method not only helps in finding rational roots but also reduces the polynomial to lower-degree factors, which can be solved using other techniques or formulas.
3. Completing the Factoring Process
Now, let’s put everything together. We started with f(x) = -2x⁵ - x⁴ + 10x³ + 5x², factored out x², and then used factoring by grouping on the remaining cubic polynomial. Our function is now factored as:
f(x) = x²(2x + 1)(-x² + 5)
To find the x-intercepts, we set each factor equal to zero and solve for x:
- x² = 0 gives us x = 0 (with multiplicity 2).
- (2x + 1) = 0 gives us x = -1/2.
- (-x² + 5) = 0 can be rewritten as x² = 5, which gives us x = ±√5.
Therefore, the x-intercepts of the function are x = 0, x = -1/2, x = √5, and x = -√5. These are the points where the graph of the function intersects the x-axis. The multiplicity of the root x = 0 affects the behavior of the graph at that point, causing it to touch the axis and turn back rather than cross it.
4. Verifying the Roots
To ensure the accuracy of our factored form and the x-intercepts we found, it’s essential to verify these roots. There are several methods to verify the roots, including substitution, graphical analysis, and numerical methods. Each approach provides a different perspective and helps confirm that our solutions are correct.
4.1. Substitution
One straightforward way to verify the roots is by substituting each value back into the original function f(x) and confirming that the result is zero. This method directly checks whether the values we found satisfy the equation f(x) = 0. Let’s perform this check for each of our roots:
- For x = 0:
- f(0) = -2(0)⁵ - (0)⁴ + 10(0)³ + 5(0)² = 0
- For x = -1/2:
- f(-1/2) = -2(-1/2)⁵ - (-1/2)⁴ + 10(-1/2)³ + 5(-1/2)²
- f(-1/2) = -2(-1/32) - (1/16) + 10(-1/8) + 5(1/4)
- f(-1/2) = 1/16 - 1/16 - 5/4 + 5/4 = 0
- For x = √5:
- f(√5) = -2(√5)⁵ - (√5)⁴ + 10(√5)³ + 5(√5)²
- f(√5) = -2(25√5) - 25 + 10(5√5) + 5(5)
- f(√5) = -50√5 - 25 + 50√5 + 25 = 0
- For x = -√5:
- f(-√5) = -2(-√5)⁵ - (-√5)⁴ + 10(-√5)³ + 5(-√5)²
- f(-√5) = -2(-25√5) - 25 + 10(-5√5) + 5(5)
- f(-√5) = 50√5 - 25 - 50√5 + 25 = 0
Since f(x) = 0 for each of these values, our roots are verified.
4.2. Graphical Analysis
A visual way to verify the x-intercepts is by graphing the function f(x). The points where the graph intersects the x-axis correspond to the real roots of the function. By plotting the graph, we can visually confirm that the x-intercepts match the roots we calculated.
Using graphing software or a graphing calculator, plot the function f(x) = -2x⁵ - x⁴ + 10x³ + 5x². Observe the points where the graph intersects the x-axis. You should see the graph crossing or touching the x-axis at approximately x = -2.236 (-√5), x = -0.5, x = 0, and x = 2.236 (√5). The behavior of the graph at x = 0 is particularly noteworthy; the graph touches the x-axis and turns back, indicating a root with even multiplicity (in this case, 2).
4.3. Numerical Methods
Numerical methods, such as using calculators or computer software to find the roots, can also be employed for verification. These methods provide numerical approximations of the roots and can be especially useful for polynomials that are difficult to factor analytically. By using a numerical solver, you can input the function f(x) and find its roots, which should closely match the values we calculated through factoring.
5. Significance of X-Intercepts
X-intercepts are crucial points in understanding the behavior of a function. They represent the values of x for which the function's output is zero, indicating where the graph crosses or touches the x-axis. These points are fundamental in various mathematical and real-world applications, providing insights into the function’s properties and behavior.
In the context of polynomial functions, x-intercepts help in determining the intervals where the function is positive or negative. This information is valuable in solving inequalities and understanding the function's range. Additionally, the x-intercepts, along with the leading coefficient, help in sketching the graph of the polynomial function, giving a visual representation of its behavior.
6. Real-World Applications
Understanding how to find x-intercepts has numerous real-world applications across various fields. In physics, for example, x-intercepts can represent equilibrium points in a system. In engineering, they might indicate critical points in a design where a system changes its behavior. In economics, x-intercepts can represent break-even points where costs equal revenue.
6.1. Physics
Consider a projectile motion problem where the height of the projectile is described by a quadratic function h(t), where t is time. The x-intercepts of this function (where h(t) = 0) represent the times at which the projectile hits the ground. Finding these intercepts helps in determining the total flight time and the range of the projectile.
6.2. Engineering
In structural engineering, the deflection of a beam under load can be modeled by a polynomial function. The x-intercepts of this function can represent the points along the beam where there is no deflection, which are critical for ensuring structural integrity. Engineers use these points to design and reinforce structures effectively.
6.3. Economics
In economics, cost and revenue functions are often used to analyze the profitability of a business. The points where the cost function equals the revenue function (i.e., where their difference is zero) are the break-even points. These points can be found by determining the x-intercepts of the function representing the difference between revenue and cost, providing crucial information for business planning and decision-making.
Conclusion
In this comprehensive guide, we have thoroughly explored the process of finding all x-intercepts of the function f(x) = -2x⁵ - x⁴ + 10x³ + 5x². By systematically factoring the polynomial, setting each factor to zero, and solving for x, we identified the x-intercepts as x = 0, x = -1/2, x = √5, and x = -√5. We then verified these roots using substitution and graphical analysis, confirming their accuracy.
The ability to find x-intercepts is a fundamental skill in mathematics, with broad applications across various scientific and engineering disciplines. This detailed explanation not only provides a step-by-step method but also emphasizes the importance of understanding the underlying concepts and their practical implications. Mastering these techniques will undoubtedly enhance your problem-solving skills and deepen your understanding of polynomial functions.
