Finding Zeros Of Polynomial Functions: A Step-by-Step Guide

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In the realm of algebra, polynomial functions hold a significant place, and understanding their zeros is crucial for various applications. Zeros, also known as roots or x-intercepts, are the values of x that make the function equal to zero. In this article, we delve into the process of finding the zeros of a polynomial function, specifically focusing on the given function $f(x) = 3x^3 - 8x^2 + 3x + 2$. We are provided with the information that $x = 1$ is a zero of this function, and our task is to identify another zero from the given options: A. $x = 2$, B. $x = -2$, C. $x = -1$, and D. $x = 3$.

Understanding Polynomial Zeros and the Factor Theorem

Before we embark on the journey of finding the other zeros, let's solidify our understanding of polynomial zeros and the factor theorem. A polynomial zero is a value of $x$ that, when substituted into the polynomial function, results in an output of zero. Graphically, these zeros correspond to the points where the polynomial function's graph intersects the x-axis.

The Factor Theorem provides a powerful connection between zeros and factors of a polynomial. It states that if $x = a$ is a zero of a polynomial function $f(x)$, then $(x - a)$ is a factor of $f(x)$. Conversely, if $(x - a)$ is a factor of $f(x)$, then $x = a$ is a zero of $f(x)$. This theorem serves as a cornerstone for our approach to finding the remaining zeros.

In our case, we are given that $x = 1$ is a zero of $f(x) = 3x^3 - 8x^2 + 3x + 2$. According to the Factor Theorem, this means that $(x - 1)$ is a factor of $f(x)$. This knowledge is our stepping stone to uncovering the other zeros.

Utilizing Polynomial Division to Find the Quotient

Since we know that $(x - 1)$ is a factor of $f(x)$, we can employ polynomial division to divide $f(x)$ by $(x - 1)$. This division will yield a quotient, which will be another polynomial of a lower degree. The zeros of this quotient will also be zeros of the original polynomial $f(x)$.

Let's perform the polynomial division:

 3x^2 - 5x - 2
x - 1 | 3x^3 - 8x^2 + 3x + 2
       -(3x^3 - 3x^2)
       -----------------
            -5x^2 + 3x
            -(-5x^2 + 5x)
            -----------------
                   -2x + 2
                   -(-2x + 2)
                   ---------
                         0

The result of the division is the quotient $3x^2 - 5x - 2$. This quadratic polynomial represents the remaining factor of $f(x)$ after we've accounted for the factor $(x - 1)$. Now, our focus shifts to finding the zeros of this quadratic quotient.

Finding the Zeros of the Quadratic Quotient

We now have the quadratic equation $3x^2 - 5x - 2 = 0$. To find its zeros, we can employ various techniques, such as factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the most straightforward approach.

Let's factor the quadratic expression:

3x2−5x−2=(3x+1)(x−2)3x^2 - 5x - 2 = (3x + 1)(x - 2)

Now, we set each factor equal to zero and solve for $x$:

3x + 1 = 0$ or $x - 2 = 0

Solving these equations, we get:

x = - rac{1}{3}$ or $x = 2

Therefore, the zeros of the quadratic quotient are $x = - rac{1}{3}$ and $x = 2$. These are also zeros of the original polynomial function $f(x)$.

Identifying Another Zero from the Options

We have successfully found the zeros of the polynomial function $f(x) = 3x^3 - 8x^2 + 3x + 2$. We were initially given that $x = 1$ is a zero, and through polynomial division and factoring, we discovered two more zeros: $x = - rac{1}{3}$ and $x = 2$.

Now, let's revisit the given options:

A. $x = 2$ B. $x = -2$ C. $x = -1$ D. $x = 3$

Comparing our findings with the options, we can clearly see that $x = 2$ is one of the zeros we found. Therefore, the correct answer is A. $x = 2$.

Conclusion

In this comprehensive exploration, we successfully identified another zero of the polynomial function $f(x) = 3x^3 - 8x^2 + 3x + 2$ given that $x = 1$ is a zero. We achieved this by leveraging the Factor Theorem, performing polynomial division, and factoring the resulting quadratic quotient. This process not only led us to the correct answer but also deepened our understanding of polynomial zeros and their relationship to factors.

The ability to find zeros of polynomial functions is a fundamental skill in algebra and calculus. It has wide-ranging applications in various fields, including engineering, physics, and economics. By mastering these techniques, you equip yourself with a powerful tool for solving a multitude of problems.

If $x=1$ is a zero of the polynomial function $f(x)=3x3-8x2+3x+2$, what is another zero of $f(x)$ among the following options?

A. $x=2$ B. $x=-2$ C. $x=-1$ D. $x=3$

Finding Zeros of Polynomial Functions A Step by Step Guide