Finding Zeros Of Polynomial Functions A Step-by-Step Guide
In this article, we will explore how to find the zeros of a polynomial function, using the example of f(x) = x³ - 9x² + 20x. We'll break down the process step-by-step, ensuring you understand the underlying concepts and can apply them to other polynomial functions.
Understanding Polynomial Functions and Zeros
Before diving into the solution, let's clarify some key terms.
A polynomial function is a function that can be expressed in the form:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ, aₙ₋₁, ..., a₁, a₀ are constants (called coefficients) and n is a non-negative integer (called the degree of the polynomial).
The zeros of a polynomial function are the values of x for which f(x) = 0. These are also known as the roots of the polynomial equation f(x) = 0. Geometrically, the zeros represent the points where the graph of the polynomial function intersects the x-axis.
Finding the zeros of a polynomial is a fundamental problem in algebra, with applications in various fields such as engineering, physics, and economics. Different techniques can be employed to find the zeros, depending on the degree and complexity of the polynomial. For linear and quadratic polynomials, we have straightforward formulas. However, for higher-degree polynomials, we often need to use factoring, synthetic division, or numerical methods.
Problem Statement
Our problem is to find the zeros of the polynomial function:
f(x) = x³ - 9x² + 20x
We are given that one zero is x = 0. This information is crucial and will help us simplify the process of finding the remaining zeros.
Step 1: Factoring out the Common Factor
Factoring is a powerful technique for finding the zeros of a polynomial. It involves expressing the polynomial as a product of simpler polynomials. In our case, we can observe that x is a common factor in all the terms of the polynomial:
f(x) = x³ - 9x² + 20x = x(x² - 9x + 20)
This factorization tells us immediately that x = 0 is a zero of the function, as given in the problem statement. Now, we need to find the zeros of the quadratic factor x² - 9x + 20.
The importance of factoring cannot be overstated. By factoring out common factors, we reduce the degree of the polynomial we need to work with, making it easier to find the zeros. In this case, we've reduced a cubic polynomial to a product of a linear factor (x) and a quadratic factor (x² - 9x + 20). This simplifies the problem significantly.
Factoring is not always straightforward, and different techniques may be required depending on the polynomial. For instance, we might use the difference of squares factorization (a² - b² = (a + b)(a - b)) or the sum/difference of cubes factorization (a³ ± b³ = (a ± b)(a² ∓ ab + b²)). However, in this case, we have a quadratic factor, which can be factored using techniques we'll discuss in the next step.
Step 2: Factoring the Quadratic
To find the zeros of the quadratic factor x² - 9x + 20, we need to factor it further. We are looking for two numbers that multiply to 20 and add up to -9. By considering the factors of 20 (1 and 20, 2 and 10, 4 and 5), we can see that -4 and -5 satisfy these conditions:
(-4) * (-5) = 20
(-4) + (-5) = -9
Therefore, we can factor the quadratic as follows:
x² - 9x + 20 = (x - 4)(x - 5)
Now, we have the complete factorization of the polynomial function:
f(x) = x(x - 4)(x - 5)
Factoring quadratics is a crucial skill in algebra. Several methods can be used, including factoring by grouping, the quadratic formula, and completing the square. In this case, we used the factoring by inspection method, which is efficient when the coefficients are relatively small and the factors are integers. The key is to identify two numbers that satisfy the conditions of multiplying to the constant term and adding to the coefficient of the linear term.
The quadratic formula is a general method for finding the roots of any quadratic equation of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
While the quadratic formula always works, it can be more time-consuming than factoring by inspection when the quadratic is easily factorable. In this case, factoring by inspection was the most efficient method.
Step 3: Finding the Zeros
To find the zeros of f(x), we set the function equal to zero and solve for x:
f(x) = x(x - 4)(x - 5) = 0
This equation holds true if any of the factors are equal to zero. Therefore, the zeros are:
x = 0
x - 4 = 0 => x = 4
x - 5 = 0 => x = 5
Thus, the zeros of the polynomial function f(x) = x³ - 9x² + 20x are 0, 4, and 5.
Understanding the relationship between factors and zeros is crucial. Each factor of the form (x - a) corresponds to a zero x = a. This is a direct consequence of the factor theorem, which states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. In other words, if a is a zero of the polynomial, then (x - a) is a factor, and vice versa.
The zeros of a polynomial tell us a lot about its graph. The real zeros correspond to the x-intercepts of the graph. The number of zeros (counting multiplicity) is equal to the degree of the polynomial. In this case, the polynomial is of degree 3, and we found three zeros: 0, 4, and 5. This means the graph of the polynomial will intersect the x-axis at these three points.
Conclusion
We have successfully found the zeros of the polynomial function f(x) = x³ - 9x² + 20x. The zeros are 0, 4, and 5. We achieved this by factoring out the common factor x and then factoring the resulting quadratic. This step-by-step approach can be applied to other polynomial functions, making it a valuable technique in algebra.
In summary, the key steps in finding the zeros of a polynomial are:
- Look for common factors and factor them out.
- Factor the remaining polynomial, using techniques such as factoring by inspection, the quadratic formula, or other methods.
- Set each factor equal to zero and solve for x to find the zeros.
By mastering these techniques, you can confidently find the zeros of a wide range of polynomial functions.
Final Answer
The zeros of the polynomial function f(x) = x³ - 9x² + 20x are 0, 4, and 5. Therefore, the correct answer is D. 0, 5, 4.
