Form A Polynomial { F(x) $}$ With Real Coefficients Having The Given Degree And Zeros.- Degree: 4- Zeros: { -4+2i$}$, { -3$}$ With Multiplicity 2Let { A $}$ Represent The Leading Coefficient. The Polynomial Is

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Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. When forming a polynomial with real coefficients and given zeros, we need to consider the properties of complex numbers and their conjugates. In this article, we will explore how to form a polynomial of degree 4 with real coefficients and given zeros.

Understanding Complex Zeros

Complex zeros are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

When a polynomial has real coefficients, complex zeros always appear in conjugate pairs. This means that if a polynomial has a complex zero, its conjugate is also a zero of the polynomial.

Given Zeros and Their Multiplicity

The given zeros are -4 + 2i and -3, with multiplicity 2. This means that -3 is a zero of the polynomial with multiplicity 2, which means that it is a zero of the polynomial twice.

Forming the Polynomial

To form the polynomial, we need to use the given zeros and their multiplicity. We will start by writing the factors corresponding to each zero.

  • For the zero -4 + 2i, the corresponding factor is (x - (-4 + 2i)) = (x + 4 - 2i).
  • For the zero -4 - 2i, the corresponding factor is (x - (-4 - 2i)) = (x + 4 + 2i).
  • For the zero -3, the corresponding factor is (x - (-3)) = (x + 3).

Since -3 is a zero with multiplicity 2, we need to include the factor (x + 3) twice in the polynomial.

Writing the Polynomial

Now that we have the factors, we can write the polynomial by multiplying them together.

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

To simplify the expression, we can multiply the factors together.

f(x) = a((x + 4)^2 - (2i)^2)(x + 3)^2

Using the difference of squares formula, we can simplify the expression further.

f(x) = a((x + 4)^2 + 4)(x + 3)^2

Expanding the squares, we get:

f(x) = a(x^2 + 8x + 16 + 4)(x^2 + 6x + 9)

Simplifying the expression, we get:

f(x) = a(x^2 + 8x + 20)(x^2 + 6x + 9)

Multiplying the expressions together, we get:

f(x) = a(x^4 + 14x^3 + 73x^2 + 156x + 180)

Determining the Leading Coefficient

The leading coefficient is the coefficient of the highest degree term in the polynomial. In this case, the highest degree term is x^4, and its coefficient is a.

Conclusion

In this article, we formed a polynomial of degree 4 with real coefficients and given zeros. We used the properties of complex numbers and their conjugates to determine the factors of the polynomial. We then multiplied the factors together to obtain the polynomial in factored form. Finally, we determined the leading coefficient of the polynomial.

Example Use Case

Suppose we want to find the value of the polynomial at x = 2. We can plug x = 2 into the polynomial and evaluate it.

f(2) = a(2^4 + 14(2)^3 + 73(2)^2 + 156(2) + 180)

Simplifying the expression, we get:

f(2) = a(16 + 56 + 292 + 312 + 180)

Combining like terms, we get:

f(2) = a(956)

Since we don't know the value of the leading coefficient a, we can't determine the exact value of the polynomial at x = 2.

Code Implementation

Here is a Python code implementation of the polynomial:

import numpy as np

def form_polynomial(a, x):
# Define the factors
factor1 = (x + 4 - 2j)
factor2 = (x + 4 + 2j)
factor3 = (x + 3)
factor4 = (x + 3)


# Multiply the factors together
polynomial = a * (factor1 * factor2) * (factor3 ** 2)

return polynomial



x = 2



a = 1



polynomial = form_polynomial(a, x)


print(polynomial)

Q: What is the difference between a complex zero and its conjugate?

A: A complex zero is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: Why do complex zeros always appear in conjugate pairs when a polynomial has real coefficients?

A: When a polynomial has real coefficients, complex zeros always appear in conjugate pairs because the coefficients of the polynomial are real numbers. This means that if a polynomial has a complex zero, its conjugate is also a zero of the polynomial.

Q: What is the multiplicity of a zero in a polynomial?

A: The multiplicity of a zero in a polynomial is the number of times that zero appears as a factor in the polynomial. For example, if a polynomial has a zero with multiplicity 2, that means that the zero appears twice as a factor in the polynomial.

Q: How do you form a polynomial with real coefficients and given zeros?

A: To form a polynomial with real coefficients and given zeros, you need to use the given zeros and their multiplicity. You will start by writing the factors corresponding to each zero, and then you will multiply the factors together to obtain the polynomial in factored form.

Q: What is the leading coefficient of a polynomial?

A: The leading coefficient of a polynomial is the coefficient of the highest degree term in the polynomial. In the case of the polynomial we formed in this article, the leading coefficient is a.

Q: How do you determine the value of a polynomial at a given value of x?

A: To determine the value of a polynomial at a given value of x, you need to plug the value of x into the polynomial and evaluate it. This will give you the value of the polynomial at that specific value of x.

Q: What is the importance of the leading coefficient in a polynomial?

A: The leading coefficient is an important part of a polynomial because it determines the degree of the polynomial and the direction of the graph of the polynomial. In the case of the polynomial we formed in this article, the leading coefficient is a, and it determines the degree of the polynomial and the direction of the graph.

Q: Can you provide an example of how to form a polynomial with real coefficients and given zeros?

A: Yes, let's say we want to form a polynomial with real coefficients and given zeros -2 + 3i, -2 - 3i, and 1 with multiplicity 2. We can start by writing the factors corresponding to each zero:

  • For the zero -2 + 3i, the corresponding factor is (x - (-2 + 3i)) = (x + 2 - 3i).
  • For the zero -2 - 3i, the corresponding factor is (x - (-2 - 3i)) = (x + 2 + 3i).
  • For the zero 1 with multiplicity 2, the corresponding factor is (x - 1)(x - 1) = (x - 1)^2.

We can then multiply the factors together to obtain the polynomial in factored form:

f(x) = a((x + 2 - 3i)(x + 2 + 3i))(x - 1)^2

Simplifying the expression, we get:

f(x) = a((x + 2)^2 - (3i)^2)(x - 1)^2

Using the difference of squares formula, we can simplify the expression further:

f(x) = a((x + 2)^2 + 9)(x - 1)^2

Expanding the squares, we get:

f(x) = a(x^2 + 4x + 4 + 9)(x^2 - 2x + 1)

Simplifying the expression, we get:

f(x) = a(x^2 + 4x + 13)(x^2 - 2x + 1)

Multiplying the expressions together, we get:

f(x) = a(x^4 + 2x^3 - 5x^2 + 6x + 13)

Q: Can you provide a code implementation of forming a polynomial with real coefficients and given zeros?

A: Yes, here is a Python code implementation of forming a polynomial with real coefficients and given zeros:

import numpy as np

def form_polynomial(a, x):
# Define the factors
factor1 = (x + 2 - 3j)
factor2 = (x + 2 + 3j)
factor3 = (x - 1)
factor4 = (x - 1)


# Multiply the factors together
polynomial = a * (factor1 * factor2) * (factor3 ** 2)

return polynomial




x = 2



a = 1



polynomial = form_polynomial(a, x)


print(polynomial)

This code defines a function form_polynomial that takes the leading coefficient a and the value of x as input and returns the polynomial evaluated at x. The code then defines the value of x and the leading coefficient a, evaluates the polynomial at x = 2, and prints the result.