Find Polynomial Roots: -5, -4, 1
Find Polynomial Roots: -5, -4, 1
Hey math enthusiasts! Ever wondered how to construct a polynomial when you're given its roots? It's like having the building blocks and knowing how to put them together to create a unique structure. Today, we're going to tackle a specific problem: what polynomial has roots of -5, -4, and 1? This isn't just an abstract math puzzle; understanding this process is fundamental to grasping the behavior of polynomial functions, their graphs, and their applications in various fields like engineering, economics, and physics. We'll break down the concept, work through the solution step-by-step, and help you feel confident in your ability to solve similar problems. So, grab your thinking caps, and let's dive into the fascinating world of polynomials!
Understanding the Relationship Between Roots and Factors
The core concept we need to understand when finding a polynomial from its roots is the Factor Theorem. In simple terms, the Factor Theorem states that if 'r' is a root of a polynomial P(x), then (x - r) is a factor of P(x). Conversely, if (x - r) is a factor of P(x), then 'r' is a root of P(x). This is a powerful tool because it allows us to move back and forth between the roots of a polynomial and its factored form. For our problem, we are given the roots: -5, -4, and 1. According to the Factor Theorem, if these are the roots, then the corresponding factors must be:
- For the root -5, the factor is (x - (-5)), which simplifies to (x + 5).
- For the root -4, the factor is (x - (-4)), which simplifies to (x + 4).
- For the root 1, the factor is (x - 1).
So, we know that our polynomial must have these three factors. A polynomial can be expressed as the product of its factors. Therefore, our polynomial, let's call it P(x), will be of the form:
P(x) = a * (x + 5) * (x + 4) * (x - 1)
where 'a' is a non-zero constant. Unless specified otherwise, or if we are given a specific point the polynomial passes through, we usually assume 'a' to be 1 for simplicity. This gives us the simplest polynomial with these roots. For the purpose of multiple-choice questions, the constant 'a' is often implicitly set to 1, as the options provided will likely reflect this.
Expanding the Factors to Find the Polynomial
Now that we have the factors, the next step is to multiply them together to get the polynomial in its standard form (ax^n + bx^(n-1) + ... + c). Since we have three linear factors, we know our resulting polynomial will be a cubic polynomial (degree 3). Let's start by multiplying two of the factors, and then multiply the result by the third factor. It doesn't matter which order we choose, but let's begin with (x + 5) and (x + 4):
(x + 5)(x + 4) = x * (x + 4) + 5 * (x + 4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20
Great! Now we have a quadratic expression. The next step is to multiply this result by the remaining factor, (x - 1):
(x^2 + 9x + 20)(x - 1) = x^2 * (x - 1) + 9x * (x - 1) + 20 * (x - 1)
Let's distribute:
= (x^3 - x^2) + (9x^2 - 9x) + (20x - 20)
Now, we combine like terms:
= x^3 + (-x^2 + 9x^2) + (-9x + 20x) - 20
= x^3 + 8x^2 + 11x - 20
So, the polynomial with roots -5, -4, and 1, assuming the leading coefficient is 1, is x^3 + 8x^2 + 11x - 20. This is the expanded form we were looking for. It's crucial to be meticulous during the expansion process, as a single arithmetic error can lead to a completely different result. Double-checking each step, especially when combining terms, is always a good practice.
Checking the Options and Confirming the Answer
We've derived our polynomial: x^3 + 8x^2 + 11x - 20. Now, let's look at the provided options to see which one matches our result.
A. B. C. D.
Our calculated polynomial, x^3 + 8x^2 + 11x - 20, perfectly matches option D. Therefore, option D is the correct answer.
It's always a good idea to have a way to verify your answer. One quick method, especially if you're unsure about the expansion, is to plug one of the roots back into the polynomial and see if it results in zero. Let's test with the root x = 1 in option D:
P(1) = (1)^3 + 8(1)^2 + 11(1) - 20 = 1 + 8 + 11 - 20 = 20 - 20 = 0
Since P(1) = 0, x = 1 is indeed a root of this polynomial. You could perform similar checks for x = -4 and x = -5 to further confirm. For example, let's check x = -4:
P(-4) = (-4)^3 + 8(-4)^2 + 11(-4) - 20 = -64 + 8(16) - 44 - 20 = -64 + 128 - 44 - 20 = 128 - 128 = 0
And for x = -5:
P(-5) = (-5)^3 + 8(-5)^2 + 11(-5) - 20 = -125 + 8(25) - 55 - 20 = -125 + 200 - 55 - 20 = 200 - 200 = 0
All three roots yield 0 when substituted into the polynomial in option D, confirming it is the correct polynomial. This verification process adds an extra layer of confidence in your solution, especially in high-stakes situations like exams.
Why This Matters: Applications of Polynomials
Understanding how to construct polynomials from their roots isn't just an academic exercise; it's a foundational skill with broad applications. Polynomials are used everywhere! In engineering, they help model the trajectory of projectiles, the stress on structures, and the design of circuits. Economics uses polynomials to forecast market trends, analyze costs, and optimize profits. Computer graphics rely heavily on polynomials to create smooth curves and animations. Even in biology, polynomials can model population growth or the spread of diseases. The ability to relate the roots (where the function crosses the x-axis, often representing critical points or zero values) to the function's equation provides deep insights into the behavior of these systems. When you see a graph that looks like a curve, chances are a polynomial is behind it, and knowing its roots tells you crucial information about its behavior. The process we just went through – taking roots, forming factors, and expanding – is the reverse of finding roots, and both sides of this coin are essential for a complete understanding. This skill empowers you to not only analyze existing models but also to create new ones to solve real-world problems. The elegance of mathematics lies in these connections, allowing us to describe complex phenomena with relatively simple algebraic expressions.
Conclusion
We successfully determined that the polynomial with roots -5, -4, and 1 is x^3 + 8x^2 + 11x - 20. This was achieved by applying the Factor Theorem, which states that if 'r' is a root, then (x - r) is a factor. We converted each root into its corresponding linear factor: (x + 5), (x + 4), and (x - 1). By multiplying these factors together, we obtained the polynomial in its expanded form. We also verified our answer by plugging the roots back into the polynomial and confirming that each resulted in zero, and by comparing our result with the given multiple-choice options. Mastering this process is key to understanding polynomial functions and their wide-ranging applications in science, technology, and economics. Keep practicing, and you'll find yourself navigating the world of polynomials with ease!
For further exploration and to deepen your understanding of polynomial functions and their properties, you can refer to resources from The Art of Problem Solving and Khan Academy. These platforms offer comprehensive explanations, practice problems, and a wealth of information on algebra and calculus topics.
