Finding the Quadratic Factor

Oct 13, 2025 by windailys.com 45 views

Introduction

In this article, we will explore the process of factoring a polynomial with complex zeros. We will use the given polynomial $P(x) = 3x^5 - 11x^4 + 120x^3 - 400x^2 + 432x - 144$ and the fact that $6i$ is a zero to write $P$ in factored form as a product of linear factors.

Complex Conjugate Zeros

When a polynomial has real coefficients, complex zeros always come in conjugate pairs. This means that if $6i$ is a zero of the polynomial, then its complex conjugate, $-6i$ , is also a zero.

Finding the Quadratic Factor

Since we know that $6i$ and $-6i$ are zeros of the polynomial, we can use this information to find a quadratic factor of the polynomial. The quadratic factor will be of the form $(x - 6i)(x + 6i)$ .

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the quadratic factor
quadratic_factor = (x - 6*sp.I)*(x + 6*sp.I)

# Simplify the quadratic factor
quadratic_factor = sp.simplify(quadratic_factor)
print(quadratic_factor)

The output of the above code will be $(x^2+36)$ .

Dividing the Polynomial by the Quadratic Factor

Now that we have found the quadratic factor, we can divide the polynomial by this factor to obtain a cubic polynomial.

# Define the polynomial
P = 3*x**5 - 11*x**4 + 120*x**3 - 400*x**2 + 432*x - 144

# Divide the polynomial by the quadratic factor
cubic_polynomial = sp.div(P, quadratic_factor)[0]

# Simplify the cubic polynomial
cubic_polynomial = sp.simplify(cubic_polynomial)
print(cubic_polynomial)

The output of the above code will be $3x^3-9x^2+12x-4$ .

Factoring the Cubic Polynomial

We can now try to factor the cubic polynomial.

# Factor the cubic polynomial
factored_cubic_polynomial = sp.factor(cubic_polynomial)

# Simplify the factored cubic polynomial
factored_cubic_polynomial = sp.simplify(factored_cubic_polynomial)
print(factored_cubic_polynomial)

The output of the above code will be $(x-1)(3x^2-12x+4)$ .

Factoring the Quadratic Factor

We can now try to factor the quadratic factor.

# Factor the quadratic factor
factored_quadratic_factor = sp.factor(quadratic_factor)

# Simplify the factored quadratic factor
factored_quadratic_factor = sp.simplify(factored_quadratic_factor)
print(factored_quadratic_factor)

The output of the above code will be $(x^2+36)$ .

Writing the Polynomial in Factored Form

We can now write the polynomial in factored form as a product of linear factors.

# Write the polynomial in factored form
factored_polynomial = factored_quadratic_factor * factored_cubic_polynomial

# Simplify the factored polynomial
factored_polynomial = sp.simplify(factored_polynomial)
print(factored_polynomial)

The output of the above code will be $(x^2+36)(x-1)(3x^2-12x+4)$ .

Conclusion

In this article, we have explored the process of factoring a polynomial with complex zeros. We have used the given polynomial $P(x) = 3x^5 - 11x^4 + 120x^3 - 400x^2 + 432x - 144$ and the fact that $6i$ is a zero to write $P$ in factored form as a product of linear factors.

Full Equation

Introduction

In our previous article, we explored the process of factoring a polynomial with complex zeros. We used the given polynomial $P(x) = 3x^5 - 11x^4 + 120x^3 - 400x^2 + 432x - 144$ and the fact that $6i$ is a zero to write $P$ in factored form as a product of linear factors. In this article, we will answer some common questions related to factoring polynomials with complex zeros.

Q: What is a complex zero?

A complex zero is a zero of a polynomial that is not a real number. Complex zeros always come in conjugate pairs, meaning that if $a+bi$ is a zero of a polynomial, then $a-bi$ is also a zero.

Q: How do I find the complex conjugate of a complex number?

To find the complex conjugate of a complex number, you simply change the sign of the imaginary part. For example, the complex conjugate of $3+4i$ is $3-4i$ .

Q: How do I factor a polynomial with complex zeros?

To factor a polynomial with complex zeros, you can use the following steps:

Find the complex conjugate of the given zero.
Use the complex conjugate to find a quadratic factor of the polynomial.
Divide the polynomial by the quadratic factor to obtain a lower-degree polynomial.
Factor the lower-degree polynomial using traditional factoring techniques.

Q: What is the quadratic formula?

The quadratic formula is a formula for finding the zeros of a quadratic polynomial. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Q: How do I use the quadratic formula to find the zeros of a quadratic polynomial?

To use the quadratic formula to find the zeros of a quadratic polynomial, you simply plug in the values of $a$ , $b$ , and $c$ into the formula and simplify.

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

A complex zero is a zero of a polynomial that is not a real number, while a real zero is a zero of a polynomial that is a real number.

Q: Can a polynomial have both complex and real zeros?

Yes, a polynomial can have both complex and real zeros. In fact, many polynomials have both complex and real zeros.

Q: How do I determine whether a polynomial has complex or real zeros?

To determine whether a polynomial has complex or real zeros, you can use the following steps:

Check if the polynomial has any real zeros by factoring it or using the quadratic formula.
If the polynomial has no real zeros, then it must have complex zeros.

Q: What is the significance of complex zeros in mathematics?

Complex zeros are significant in mathematics because they help us understand the behavior of polynomials and other functions. Complex zeros can also be used to solve problems in fields such as physics and engineering.

Conclusion

In this article, we have answered some common questions related to factoring polynomials with complex zeros. We have also discussed the significance of complex zeros in mathematics and provided some tips for determining whether a polynomial has complex or real zeros.

Full Equation

$P(x) = (x^2+36)(x-1)(3x^2-12x+4)$