Given P ( X ) = X 3 − 3 X 2 + 4 X − 12 P(x) = X^3 - 3x^2 + 4x - 12 P ( X ) = X 3 − 3 X 2 + 4 X − 12 , Find The Zeros, Real And Non-real, Of P P P .The Zeros Are: □ \square □ Write P P P In Factored Form (as A Product Of Linear Factors). P ( X ) = □ P(x) = \square P ( X ) = □

Mar 9, 2025 by ADMIN 276 views

**Finding the Zeros of a Cubic Polynomial: A Step-by-Step Guide**

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Introduction

In algebra, finding the zeros of a polynomial is a crucial concept that helps us understand the behavior of the function. A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In this article, we will focus on finding the zeros of a cubic polynomial, specifically the polynomial $P(x) = x^3 - 3x^2 + 4x - 12$ . We will use various techniques, including factoring and the Rational Root Theorem, to find the zeros of $P$ .

The Rational Root Theorem

The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ , where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term $a_0$ , and $q$ must be a factor of the leading coefficient $a_n$ . In our case, the constant term is $-12$ , and the leading coefficient is $1$ . Therefore, the possible rational roots of $P$ are the factors of $-12$ , which are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$ .

Factoring the Polynomial

To find the zeros of $P$ , we can try to factor the polynomial. We can start by looking for a common factor. In this case, we can factor out a $-1$ from the polynomial:

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

Now, we can try to factor the polynomial inside the parentheses. We can start by looking for two numbers whose product is $-12$ and whose sum is $-3$ . These numbers are $-4$ and $3$ , so we can write:

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

Now, we can factor out the common factor $(x - 3)$ :

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

Finding the Zeros

Now that we have factored the polynomial, we can find the zeros by setting each factor equal to zero. We have two factors: $(x - 3)$ and $(x^2 + 4)$ . Setting the first factor equal to zero, we get:

x - 3 = 0 \Rightarrow x = 3

This is one of the zeros of $P$ . To find the other zeros, we need to solve the equation $(x^2 + 4) = 0$ . However, this equation has no real solutions, since the square of any real number is non-negative, and adding $4$ to it will always result in a positive number. Therefore, the other zeros of $P$ are non-real.

The Zeros of $P$

In conclusion, the zeros of the polynomial $P(x) = x^3 - 3x^2 + 4x - 12$ are:

$x = 3$ (real zero)
$x = \pm i\sqrt{4}$ (non-real zeros)

We can write $P$ in factored form as:

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

Discussion

Finding the zeros of a polynomial is an important concept in algebra, and it has many applications in various fields, such as physics, engineering, and economics. In this article, we used the Rational Root Theorem and factoring to find the zeros of a cubic polynomial. We also discussed the importance of understanding the behavior of a function, and how finding the zeros can help us do that.

Conclusion

In conclusion, finding the zeros of a polynomial is a crucial concept in algebra, and it has many applications in various fields. By using the Rational Root Theorem and factoring, we can find the zeros of a polynomial, and understand the behavior of the function. We hope that this article has provided a clear and concise explanation of how to find the zeros of a polynomial, and we encourage readers to practice finding the zeros of different polynomials.

References

Glossary

Rational Root Theorem: A theorem that states that if a rational number $p/q$ is a root of the polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ , then $p$ must be a factor of the constant term $a_0$ , and $q$ must be a factor of the leading coefficient $a_n$ .
Factoring: The process of expressing a polynomial as a product of simpler polynomials.
Zeros: The values of the variable that make the polynomial equal to zero.

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Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem that states that if a rational number $p/q$ is a root of the polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ , then $p$ must be a factor of the constant term $a_0$ , and $q$ must be a factor of the leading coefficient $a_n$ .

Q: How do I find the zeros of a polynomial using the Rational Root Theorem?

A: To find the zeros of a polynomial using the Rational Root Theorem, you need to follow these steps:

List all the possible rational roots of the polynomial by finding the factors of the constant term and the leading coefficient.
Test each possible rational root by substituting it into the polynomial and checking if it equals zero.
If a rational root is found, you can use synthetic division or polynomial long division to factor the polynomial and find the other zeros.

Q: What is factoring, and how do I use it to find the zeros of a polynomial?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. To use factoring to find the zeros of a polynomial, you need to follow these steps:

Look for a common factor in the polynomial.
Factor out the common factor.
Use the factored form of the polynomial to find the zeros by setting each factor equal to zero.

Q: What are the zeros of a polynomial, and why are they important?

A: The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. The zeros of a polynomial are important because they help us understand the behavior of the function. By finding the zeros of a polynomial, we can determine the intervals where the function is increasing or decreasing, and we can also use the zeros to graph the function.

Q: How do I find the zeros of a polynomial with complex roots?

A: To find the zeros of a polynomial with complex roots, you need to use the quadratic formula or the factored form of the polynomial. If the polynomial has complex roots, you can use the factored form to find the zeros by setting each factor equal to zero. The complex roots will be in the form of $a \pm bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

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

A: A real zero is a zero that is a real number, while a non-real zero is a zero that is a complex number. Real zeros are the values of the variable that make the polynomial equal to zero, while non-real zeros are the values of the variable that make the polynomial equal to zero, but are not real numbers.

Q: How do I use the zeros of a polynomial to graph the function?

A: To use the zeros of a polynomial to graph the function, you need to follow these steps:

Plot the zeros of the polynomial on a number line.
Use the zeros to determine the intervals where the function is increasing or decreasing.
Plot the function by drawing a smooth curve that passes through the zeros and the points where the function is increasing or decreasing.

Q: What are some common mistakes to avoid when finding the zeros of a polynomial?

A: Some common mistakes to avoid when finding the zeros of a polynomial include:

Not listing all the possible rational roots of the polynomial.
Not testing each possible rational root by substituting it into the polynomial.
Not using synthetic division or polynomial long division to factor the polynomial.
Not setting each factor equal to zero to find the zeros.
Not using the zeros to determine the intervals where the function is increasing or decreasing.

Q: How do I check my work when finding the zeros of a polynomial?

A: To check your work when finding the zeros of a polynomial, you need to follow these steps:

Plug the zeros back into the polynomial to check if they equal zero.
Use synthetic division or polynomial long division to factor the polynomial and check if the zeros are correct.
Use the factored form of the polynomial to find the zeros and check if they are correct.

Q: What are some real-world applications of finding the zeros of a polynomial?

A: Some real-world applications of finding the zeros of a polynomial include:

Physics: Finding the zeros of a polynomial can help us understand the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
Engineering: Finding the zeros of a polynomial can help us design and optimize systems, such as electronic circuits or mechanical systems.
Economics: Finding the zeros of a polynomial can help us understand the behavior of economic systems, such as the behavior of supply and demand curves.

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

A: To use technology to find the zeros of a polynomial, you can use a graphing calculator or a computer algebra system (CAS). These tools can help you find the zeros of a polynomial by plotting the function and finding the points where the function intersects the x-axis.

Q: What are some common tools used to find the zeros of a polynomial?

A: Some common tools used to find the zeros of a polynomial include:

Graphing calculators
Computer algebra systems (CAS)
Synthetic division
Polynomial long division
Factoring

Q: How do I choose the right tool for finding the zeros of a polynomial?

A: To choose the right tool for finding the zeros of a polynomial, you need to consider the following factors:

The complexity of the polynomial
The number of zeros you need to find
The level of accuracy you need
The tools available to you

By considering these factors, you can choose the right tool for finding the zeros of a polynomial and make the process easier and more efficient.