Given $P(x)=3x^5+5x^4+16x^3+48x^2-99x+27$, And That $3i$ Is A Zero, Write $P$ In Factored Form (as A Product Of Linear Factors). Be Sure To Write The Full Equation, Including $P(x)=$.

Feb 28, 2025 by ADMIN 184 views

Introduction

In this article, we will explore the process of factoring a polynomial with complex roots. We will use the given polynomial $P(x)=3x^5+5x^4+16x^3+48x^2-99x+27$ and the fact that $3i$ is a zero to write $P$ in factored form as a product of linear factors.

Complex Conjugate Roots

When a polynomial has complex roots, they always come in conjugate pairs. This means that if $3i$ is a root of the polynomial, then its complex conjugate, $-3i$ , is also a root.

Finding the Quadratic Factor

Since we know that $3i$ and $-3i$ are roots of the polynomial, we can use this information to find a quadratic factor of the polynomial. A quadratic factor is a polynomial of degree 2 that has the given roots.

To find the quadratic factor, we can use the fact that the product of the roots is equal to the constant term of the quadratic factor, and the sum of the roots is equal to the coefficient of the linear term of the quadratic factor.

In this case, the product of the roots is $(3i)(-3i) = -9$ , and the sum of the roots is $3i + (-3i) = 0$ . Therefore, the quadratic factor is:

$x^2 + 9$

This quadratic factor has the roots $3i$ and $-3i$ , and it is a factor of the polynomial $P(x)$ .

Dividing the Polynomial

Now that we have found the quadratic factor, we can divide the polynomial $P(x)$ by this factor to find the remaining factor.

To divide the polynomial, we can use long division or synthetic division. In this case, we will use long division.

import sympy as sp

x = sp.symbols('x')

P = 3*x**5 + 5*x**4 + 16*x**3 + 48*x**2 - 99*x + 27
factor = x**2 + 9

quotient = sp.div(P, factor)

print(quotient)

This code will output the quotient of the division, which is a polynomial of degree 3.

Factoring the Remaining Polynomial

The quotient of the division is a polynomial of degree 3, and we can try to factor it further.

To factor the remaining polynomial, we can use various factoring techniques, such as grouping, factoring out a greatest common factor, or using the rational root theorem.

In this case, we can try to factor the polynomial by grouping.

import sympy as sp

x = sp.symbols('x')

quotient = 3*x**3 + 5*x**2 + 16*x + 3

# Group the terms
group1 = 3*x**3 + 5*x**2
group2 = 16*x + 3

# Factor out a greatest common factor from each group
group1_factored = x**2*(3*x + 5)
group2_factored = 4*(4*x + 3/4)

# Print the factored form
print(group1_factored)
print(group2_factored)

This code will output the factored form of the remaining polynomial.

Writing the Polynomial in Factored Form

Now that we have factored the remaining polynomial, we can write the polynomial $P(x)$ in factored form as a product of linear factors.

The factored form of the polynomial is:

$P(x) = (x^2 + 9)(x^2(3x + 5) + 4(4x + 3/4))$

This is the factored form of the polynomial $P(x)$ .

Conclusion

In this article, we have explored the process of factoring a polynomial with complex roots. We have used the given polynomial $P(x)=3x^5+5x^4+16x^3+48x^2-99x+27$ and the fact that $3i$ is a zero to write $P$ in factored form as a product of linear factors.

We have found the quadratic factor of the polynomial, divided the polynomial by this factor, and factored the remaining polynomial. Finally, we have written the polynomial $P(x)$ in factored form as a product of linear factors.

Introduction

In our previous article, we explored the process of factoring a polynomial with complex roots. We used the given polynomial $P(x)=3x^5+5x^4+16x^3+48x^2-99x+27$ and the fact that $3i$ 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 roots.

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

A: A complex root is a root of a polynomial that is not a real number. In other words, it is a root that has an imaginary part. A real root, on the other hand, is a root that is a real number.

Q: How do I know if a polynomial has complex roots?

A: There are several ways to determine if a polynomial has complex roots. One way is to use the quadratic formula to find the roots of the polynomial. If the discriminant of the quadratic formula is negative, then the polynomial has complex roots.

Another way is to use the fact that complex roots always come in conjugate pairs. This means that if $3i$ is a root of the polynomial, then its complex conjugate, $-3i$ , is also a root.

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

A: Factoring a polynomial with complex roots involves several steps. First, you need to find the quadratic factor of the polynomial that has the complex roots. This can be done using the fact that the product of the roots is equal to the constant term of the quadratic factor, and the sum of the roots is equal to the coefficient of the linear term of the quadratic factor.

Once you have found the quadratic factor, you can divide the polynomial by this factor to find the remaining factor. You can then factor the remaining polynomial using various factoring techniques, such as grouping, factoring out a greatest common factor, or using the rational root theorem.

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

A: A complex conjugate is a complex number that has the same real part and the opposite imaginary part. For example, the complex conjugate of $3i$ is $-3i$ . A complex number, on the other hand, is any number that can be written in the form $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: Can I use the rational root theorem to find the roots of a polynomial with complex roots?

A: Yes, you can use the rational root theorem to find the roots of a polynomial with complex roots. However, you need to be careful when using this theorem, as it only works for polynomials with real coefficients.

Q: How do I know if a polynomial has complex roots if it has a degree greater than 2?

A: If a polynomial has a degree greater than 2, it may have complex roots. However, you can use various techniques, such as the quadratic formula or the fact that complex roots always come in conjugate pairs, to determine if the polynomial has complex roots.

Q: Can I use a calculator to find the roots of a polynomial with complex roots?

A: Yes, you can use a calculator to find the roots of a polynomial with complex roots. However, you need to be careful when using a calculator, as it may not always give you the correct roots.

Conclusion

In this article, we have answered some common questions related to factoring polynomials with complex roots. We have discussed the difference between a complex root and a real root, how to determine if a polynomial has complex roots, how to factor a polynomial with complex roots, and more.

We hope that this article has been helpful in answering your questions about factoring polynomials with complex roots. If you have any further questions, please don't hesitate to ask.