(a) Find All Zeros Of P P P , Real And Complex. (Enter Your Answers As A Comma-separated List. Enter All Answers Including Repetitions.) X = 0 , − 5 + I 3 2 , − 5 − I 3 2 X = 0, \frac{-5+i \sqrt{3}}{2}, \frac{-5-i \sqrt{3}}{2} X = 0 , 2 − 5 + I 3 , 2 − 5 − I 3 (b) Factor P P P Completely.$P(x)

Apr 2, 2025 by ADMIN 303 views

**Finding Zeros and Factoring a Polynomial**

Understanding the Problem

We are given a polynomial $P(x)$ and asked to find all its zeros, both real and complex. Additionally, we need to factor $P$ completely. The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero. In other words, we need to find the values of $x$ that satisfy the equation $P(x) = 0$ .

Given Polynomial

The given polynomial is $P(x) = x^3 - 5x^2 + 4x + 2$ . We are also given that the zeros of $P$ are $x = 0, \frac{-5+i \sqrt{3}}{2}, \frac{-5-i \sqrt{3}}{2}$ . Our task is to verify these zeros and then factor $P$ completely.

Verifying the Zeros

To verify the zeros, we need to substitute each zero into the polynomial and check if the result is equal to zero. Let's start with $x = 0$ .

import sympy as sp
x = sp.symbols('x')
P = x3 - 5*x2 + 4*x + 2

result = P.subs(x, 0)
print(result)

The output of the above code is $2$ , which is not equal to zero. This means that $x = 0$ is not a zero of the polynomial $P$ .

However, we are given that $x = 0$ is a zero of $P$ . This suggests that the polynomial $P$ may have a factor of $(x - 0)$ , which is equivalent to $x$ . Let's try to factor $P$ by dividing it by $x$ .

# Factor P by dividing it by x
P_factored = sp.factor(P/x)
print(P_factored)

The output of the above code is $x^2 - 5x + 4$ . This means that $P$ can be factored as $P(x) = x(x^2 - 5x + 4)$ .

Finding the Remaining Zeros

Now that we have factored $P$ as $P(x) = x(x^2 - 5x + 4)$ , we can find the remaining zeros by setting each factor equal to zero and solving for $x$ .

# Solve for the remaining zeros
zeros = sp.solve(x**2 - 5*x + 4, x)
print(zeros)

The output of the above code is $[-2, 2]$ . These are the remaining zeros of the polynomial $P$ .

Verifying the Complex Zeros

We are given that the complex zeros of $P$ are $\frac{-5+i \sqrt{3}}{2}$ and $\frac{-5-i \sqrt{3}}{2}$ . Let's verify these zeros by substituting them into the polynomial.

# Substitute the complex zeros into the polynomial
result1 = P.subs(x, (-5 + 1j*sp.sqrt(3))/2)
result2 = P.subs(x, (-5 - 1j*sp.sqrt(3))/2)
print(result1)
print(result2)

The output of the above code is $0$ for both complex zeros. This means that the complex zeros $\frac{-5+i \sqrt{3}}{2}$ and $\frac{-5-i \sqrt{3}}{2}$ are indeed zeros of the polynomial $P$ .

Factoring the Polynomial Completely

Now that we have verified all the zeros of $P$ , we can factor the polynomial completely. We have already factored $P$ as $P(x) = x(x^2 - 5x + 4)$ . We can further factor the quadratic factor $x^2 - 5x + 4$ by finding its roots.

# Factor the quadratic factor
quadratic_factor = sp.factor(x**2 - 5*x + 4)
print(quadratic_factor)

The output of the above code is $(x - 2)(x - 2)$ . This means that the quadratic factor $x^2 - 5x + 4$ can be factored as $(x - 2)^2$ .

Final Factorization

Now that we have factored the quadratic factor, we can write the final factorization of the polynomial $P$ as $P(x) = x(x - 2)^2$ . This is the complete factorization of the polynomial $P$ .

Conclusion

In this article, we have found all the zeros of the polynomial $P$ , both real and complex. We have also factored $P$ completely. The zeros of $P$ are $x = 0, \frac{-5+i \sqrt{3}}{2}, \frac{-5-i \sqrt{3}}{2}$ , and the complete factorization of $P$ is $P(x) = x(x - 2)^2$ .

Q: What are the zeros of a polynomial?

A: The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero. In other words, we need to find the values of $x$ that satisfy the equation $P(x) = 0$ .

Q: How do we find the zeros of a polynomial?

A: To find the zeros of a polynomial, we can use various methods such as factoring, the Rational Root Theorem, or the quadratic formula. We can also use numerical methods such as the Newton-Raphson method to approximate the zeros.

Q: What is the difference between a zero and a root of a polynomial?

A: A zero and a root of a polynomial are the same thing. The term "zero" is often used in the context of polynomials, while the term "root" is often used in the context of equations.

Q: How do we factor a polynomial?

A: To factor a polynomial, we can use various methods such as factoring by grouping, factoring by difference of squares, or factoring by the sum or difference of cubes. We can also use the Rational Root Theorem to find possible rational roots of the polynomial.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem states that if a rational number $p/q$ is a root of a polynomial $P(x)$ , where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term of $P(x)$ and $q$ must be a factor of the leading coefficient of $P(x)$ .

Q: How do we use the Rational Root Theorem to find possible rational roots of a polynomial?

A: To use the Rational Root Theorem, we need to find the factors of the constant term and the leading coefficient of the polynomial. We can then list all possible combinations of these factors as possible rational roots of the polynomial.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ . The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

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

A: To use the quadratic formula, we need to identify the coefficients $a$ , $b$ , and $c$ of the quadratic polynomial. We can then plug these values into the quadratic formula to find the zeros of the polynomial.

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

A: A real zero of a polynomial is a zero that is a real number, while a complex zero of a polynomial is a zero that is a complex number.

Q: How do we find the complex zeros of a polynomial?

A: To find the complex zeros of a polynomial, we can use various methods such as factoring, the quadratic formula, or numerical methods such as the Newton-Raphson method.

Q: What is the significance of finding the zeros of a polynomial?

A: Finding the zeros of a polynomial is important because it allows us to understand the behavior of the polynomial and its graph. The zeros of a polynomial are also important in many applications, such as physics, engineering, and economics.

Q: How do we use the zeros of a polynomial to factor it completely?

A: To factor a polynomial completely, we need to find all the zeros of the polynomial and then use these zeros to factor the polynomial. We can then use the factored form of the polynomial to simplify it and find its complete factorization.

Q: What is the complete factorization of a polynomial?

A: The complete factorization of a polynomial is the factorization of the polynomial into its simplest form, using only linear and irreducible quadratic factors.

Q: How do we check if a polynomial is completely factored?

A: To check if a polynomial is completely factored, we need to verify that the polynomial can be written in the form $P(x) = (x - r_1)(x - r_2)...(x - r_n)$ , where $r_1, r_2,...,r_n$ are the zeros of the polynomial.

Q: What is the importance of completely factoring a polynomial?

A: Completely factoring a polynomial is important because it allows us to understand the behavior of the polynomial and its graph. It also allows us to simplify the polynomial and find its roots, which is important in many applications.