Polynomial Function With Zeros -4 & 6: How To Find It

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Hey guys! Let's dive into the fascinating world of polynomials and figure out how to construct a polynomial function when we know its zeros. Specifically, we're going to tackle the question: How do we find a polynomial function that has zeros at -4 and 6? This is a common problem in algebra, and understanding the process is super useful. So, let’s break it down step by step!

Understanding Zeros and Polynomials

Before we jump into the solution, let's make sure we're all on the same page about what zeros and polynomials are. In the context of functions, zeros, also known as roots or x-intercepts, are the values of x that make the function equal to zero. For a polynomial, these zeros are where the graph of the polynomial intersects the x-axis.

Now, what exactly is a polynomial? A polynomial is an expression consisting of variables (usually denoted as x) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A general form of a polynomial function can be written as:

f(x) = a_n*x^n + a_{n-1}*x^{n-1} + ... + a_1*x + a_0

Where:

  • f(x) represents the polynomial function.
  • x is the variable.
  • a_n, a_{n-1}, ..., a_1, a_0 are the coefficients (constants).
  • n is a non-negative integer representing the degree of the polynomial.

For example, f(x) = 3x^2 - 2x + 1 is a polynomial function of degree 2 (a quadratic function), and g(x) = x^3 + 5x - 7 is a polynomial function of degree 3 (a cubic function).

The Fundamental Theorem of Algebra tells us that a polynomial of degree n will have n complex roots (zeros), counting multiplicity. This means a quadratic equation (degree 2) will have two roots, a cubic equation (degree 3) will have three roots, and so on.

Understanding this relationship between zeros and polynomial functions is the key to solving our problem. We're going to use the zeros to work backward and construct the polynomial.

Constructing the Polynomial Function

Okay, so we know our zeros are -4 and 6. Here’s how we can use this information to build our polynomial function:

1. Factors from Zeros

The first step is to convert the zeros into factors. A factor is an expression that, when multiplied by another expression, gives the polynomial. If a number c is a zero of a polynomial, then (x - c) is a factor of that polynomial. This is a crucial concept.

Given our zeros of -4 and 6, we can form the factors:

  • For the zero -4, the factor is (x - (-4)), which simplifies to (x + 4).
  • For the zero 6, the factor is (x - 6).

So, we have two factors: (x + 4) and (x - 6). These are the building blocks of our polynomial.

2. Multiplying the Factors

Now that we have the factors, the next step is to multiply them together. When we multiply the factors, we're essentially reversing the process of factoring a polynomial. By multiplying these factors, we reconstruct the polynomial function that has the given zeros.

Let's multiply (x + 4) and (x - 6):

(x + 4)(x - 6) = x(x - 6) + 4(x - 6)

Using the distributive property, we get:

= x^2 - 6x + 4x - 24

Combine the like terms:

= x^2 - 2x - 24

So, the polynomial function is f(x) = x^2 - 2x - 24. This is a quadratic polynomial (degree 2) as expected, since we had two zeros.

3. Verifying the Zeros

To make sure we did everything correctly, it's always a good idea to verify that our polynomial actually has the zeros we started with. We can do this by plugging the zeros back into the polynomial and checking if the result is zero.

Let's check for x = -4:

f(-4) = (-4)^2 - 2(-4) - 24 = 16 + 8 - 24 = 24 - 24 = 0

Great! -4 is indeed a zero of our polynomial.

Now, let's check for x = 6:

f(6) = (6)^2 - 2(6) - 24 = 36 - 12 - 24 = 36 - 36 = 0

Awesome! 6 is also a zero of our polynomial. This confirms that our polynomial function is correct.

4. General Form and Leading Coefficient

We've found one polynomial function with the zeros -4 and 6, but there are actually infinitely many! Why? Because we can multiply the entire polynomial by any non-zero constant and the zeros will remain the same.

For example, 2(x^2 - 2x - 24) or -5(x^2 - 2x - 24) would also have the same zeros. This constant is often referred to as the leading coefficient.

So, the general form of a polynomial function with zeros -4 and 6 can be written as:

f(x) = a(x^2 - 2x - 24)

Where a is any non-zero constant. If you're given an additional point that the polynomial must pass through, you can determine the specific value of a. If not, the simplest form is when a = 1, which gives us our original polynomial, f(x) = x^2 - 2x - 24.

Expanding to Higher Degree Polynomials

The process we've used here can be extended to polynomials with more zeros. If you have three zeros, you'll have three factors to multiply, resulting in a cubic polynomial (degree 3). If you have four zeros, you'll have four factors, and so on.

The key steps remain the same:

  1. Convert each zero into a factor.
  2. Multiply all the factors together.
  3. Verify the zeros (if necessary).
  4. Consider the general form with a leading coefficient.

For instance, if you wanted to find a polynomial with zeros -1, 2, and 3, you would form the factors (x + 1), (x - 2), and (x - 3), multiply them together, and you'd have a cubic polynomial.

Practical Applications and Importance

Understanding how to construct polynomial functions from their zeros isn't just a theoretical exercise. It has many practical applications in various fields, including:

  • Engineering: Polynomials are used to model various physical systems and design structures.
  • Computer Graphics: Polynomials are used to create curves and surfaces in computer-aided design (CAD) and computer graphics.
  • Economics: Polynomial functions can model cost, revenue, and profit functions.
  • Data Analysis: Polynomial regression is used to fit curves to data points.

The ability to manipulate and understand polynomials is a fundamental skill in mathematics and a valuable tool in many real-world scenarios.

Let's Summarize

To recap, finding a polynomial function with given zeros involves a few key steps:

  1. Convert Zeros to Factors: If c is a zero, then (x - c) is a factor.
  2. Multiply Factors: Multiply the factors together to obtain the polynomial.
  3. Verify Zeros: Plug the original zeros back into the polynomial to confirm they result in zero.
  4. General Form: Remember the general form f(x) = a * (polynomial), where a is the leading coefficient.

In our specific example, with zeros -4 and 6, we found the polynomial function f(x) = x^2 - 2x - 24.

Final Thoughts

Guys, I hope this breakdown has clarified how to find a polynomial function when you're given its zeros. It's a fundamental concept in algebra that builds a strong foundation for more advanced topics. Keep practicing, and you'll become a polynomial pro in no time! If you have any questions or want to explore more examples, feel free to ask. Happy problem-solving!