N 3 and 2i Are Zeros F 1 Calculator

This calculator helps determine the polynomial function f(x) when given that n=-3 and 2i are zeros. It calculates the function value at x=1 and provides a visual representation of the polynomial.

Introduction

When solving polynomial equations, knowing the zeros (roots) of the function can help determine the complete polynomial. If we know two zeros of a quadratic function, we can find the polynomial in the form f(x) = a(x - r₁)(x - r₂).

In this case, we're given that n=-3 and 2i are zeros of the function. This means the polynomial has roots at x=-3 and x=2i. We'll use this information to determine the polynomial function and evaluate it at x=1.

How to Use This Calculator

Enter the value for n (default is -3)
Enter the complex zero (default is 2i)
Click "Calculate" to see the polynomial function and its value at x=1
Review the chart showing the polynomial behavior

Formula

Given zeros at x = r₁ and x = r₂, the quadratic polynomial can be expressed as:

f(x) = a(x - r₁)(x - r₂)

For our specific case with r₁ = n = -3 and r₂ = 2i:

f(x) = a(x + 3)(x - 2i)

To find the value at x=1:

f(1) = a(1 + 3)(1 - 2i) = a(4)(1 - 2i) = 4a(1 - 2i)

For simplicity, we'll assume a=1 unless specified otherwise.

Worked Example

Let's calculate the polynomial function when n=-3 and the zero is 2i:

Given zeros: x = -3 and x = 2i
Assume a=1 for simplicity
Polynomial: f(x) = (x + 3)(x - 2i)
Expand: f(x) = x² - 2ix + 3x - 6i = x² + (3 - 2i)x - 6i
Evaluate at x=1: f(1) = 1 + (3 - 2i) - 6i = 4 - 8i

This matches our formula calculation with a=1.

Interpreting Results

The calculator provides:

The polynomial function in standard form
The value of the function at x=1
A visual chart showing the polynomial behavior

The complex nature of the zero (2i) means the polynomial will have complex coefficients. The chart helps visualize how the function behaves around the zeros.

FAQ

What if I want to use a different value for 'a'?

The calculator assumes a=1 by default. You can multiply the resulting polynomial by your desired 'a' value to adjust the scaling factor.

Can I use this calculator for higher-degree polynomials?

This calculator specifically handles quadratic polynomials (degree 2) with complex zeros. For higher-degree polynomials, you would need a more advanced calculator.

What does the chart represent?

The chart shows the real part of the polynomial function over a range of x values. It helps visualize where the function crosses zero (the roots) and how it behaves around those points.