Finding Zeros Of Polynomial Functions Factoring And Division Techniques
Polynomial functions, the cornerstone of algebraic expressions, hold a world of mathematical intricacies within their curves and coefficients. Among the most intriguing aspects of these functions are their zeros, the points where the function intersects the x-axis, signifying the values of x that render the function equal to zero. In this comprehensive exploration, we embark on a journey to dissect the polynomial function f(x) = 3x³ - 8x² + 3x + 2, armed with the knowledge that x = 1 is one of its zeros. Our mission is to unearth the other elusive zeros of this cubic equation, employing a blend of algebraic techniques and insightful deductions.
The Zero Product Property: A Foundation for Discovery
The bedrock of our quest lies in the Zero Product Property, a fundamental principle that states: if the product of two or more factors equals zero, then at least one of the factors must be zero. This property serves as our guiding light, allowing us to transform the polynomial into a product of factors, thereby revealing its zeros. Given that x = 1 is a zero of f(x), we can confidently assert that (x - 1) is a factor of the polynomial. This realization forms the cornerstone of our factorization strategy.
Polynomial Long Division: A Systematic Approach to Factorization
To unveil the remaining factors, we turn to the elegant technique of polynomial long division. By dividing f(x) by (x - 1), we systematically extract the quotient, a polynomial of lower degree that holds the key to the remaining zeros. The process unfolds as follows:
3x² - 5x - 2
x - 1 | 3x³ - 8x² + 3x + 2
-(3x³ - 3x²)
-------------
-5x² + 3x
-(-5x² + 5x)
-------------
-2x + 2
-(-2x + 2)
----------
0
The result of this division reveals that:
f(x) = (x - 1)(3x² - 5x - 2)
We have successfully decomposed the cubic polynomial into a linear factor (x - 1) and a quadratic factor (3x² - 5x - 2). The linear factor has already yielded the zero x = 1. Our next step is to conquer the quadratic factor and extract its zeros.
Factoring the Quadratic: Unveiling the Remaining Zeros
The quadratic factor, 3x² - 5x - 2, presents a familiar challenge in algebra. We seek two numbers that multiply to give the product of the leading coefficient (3) and the constant term (-2), which is -6, and add up to the middle coefficient (-5). These numbers are -6 and 1. With these numbers in hand, we rewrite the middle term and proceed with factoring by grouping:
3x² - 5x - 2 = 3x² - 6x + x - 2
= 3x(x - 2) + 1(x - 2)
= (3x + 1)(x - 2)
We have successfully factored the quadratic, obtaining:
3x² - 5x - 2 = (3x + 1)(x - 2)
The Complete Factorization: A Gateway to All Zeros
Now, we assemble the complete factorization of f(x):
f(x) = (x - 1)(3x + 1)(x - 2)
Equating each factor to zero, we unveil all the zeros of the polynomial:
x - 1 = 0 => x = 1 3x + 1 = 0 => x = -1/3 x - 2 = 0 => x = 2
Thus, the zeros of f(x) are x = 1, x = -1/3, and x = 2.
Identifying the Other Zero: A Triumph of Algebraic Deduction
Having meticulously dissected the polynomial, we now stand at the precipice of our goal. The question posed challenges us to identify another zero of f(x) from a set of options. Our analysis has revealed that the zeros of f(x) are 1, -1/3, and 2. Examining the options, we find that x = 2 aligns perfectly with our findings. Therefore, the answer is unequivocally:
A. x = 2
In this comprehensive exploration, we have delved into the intricacies of polynomial functions, wielding the Zero Product Property and polynomial long division as our tools. Through systematic factorization, we have successfully unearthed the zeros of f(x) = 3x³ - 8x² + 3x + 2, demonstrating the power of algebraic techniques in unraveling mathematical mysteries. The journey from a given zero to the complete set of zeros exemplifies the elegance and interconnectedness of mathematical concepts.
Decoding Polynomial Zeros: Finding Roots Beyond x = 1 in f(x) = 3x³ - 8x² + 3x + 2
Polynomial functions are fundamental building blocks in algebra, known for their smooth curves and predictable behavior. The zeros of a polynomial, also known as roots, are the values of x for which the function evaluates to zero. These zeros represent the points where the graph of the polynomial intersects the x-axis. Finding these zeros is a core problem in algebra with wide-ranging applications in various fields such as engineering, physics, and economics. This article delves into the process of finding the zeros of the polynomial function f(x) = 3x³ - 8x² + 3x + 2, given that x = 1 is one of its zeros. We'll explore how to utilize this information to uncover the remaining zeros of the function.
Leveraging the Factor Theorem: A Key to Unlocking Zeros
The Factor Theorem is a powerful tool that connects the zeros of a polynomial with its factors. It states that if x = a is a zero of a polynomial 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 provides a direct link between zeros and factors, allowing us to use one to find the other. Since we are given that x = 1 is a zero of f(x) = 3x³ - 8x² + 3x + 2, we can immediately deduce that (x - 1) is a factor of f(x). This knowledge is the first step in our journey to uncover the remaining zeros.
Synthetic Division: A Shortcut to Polynomial Division
While polynomial long division is a reliable method for dividing polynomials, synthetic division offers a more streamlined approach, especially when dividing by a linear factor of the form (x - a). Synthetic division is a simplified algorithm that uses only the coefficients of the polynomial and the value of a to perform the division. In our case, we want to divide f(x) = 3x³ - 8x² + 3x + 2 by (x - 1). Using synthetic division, we set up the following tableau:
1 | 3 -8 3 2
|
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We bring down the leading coefficient (3) and multiply it by 1, placing the result under the next coefficient (-8). We then add these two numbers and repeat the process until we reach the end of the row:
1 | 3 -8 3 2
| 3 -5 -2
-----------
3 -5 -2 0
The last number in the bottom row (0) represents the remainder, and the other numbers (3, -5, -2) represent the coefficients of the quotient. Since the remainder is 0, we confirm that (x - 1) is indeed a factor of f(x). The quotient is a quadratic polynomial, 3x² - 5x - 2, which is one degree lower than the original cubic polynomial. This result allows us to rewrite f(x) as:
f(x) = (x - 1)(3x² - 5x - 2)
Now, our task is to find the zeros of the quadratic factor, 3x² - 5x - 2.
Quadratic Formula: A Universal Solver for Quadratic Equations
For quadratic equations of the form ax² + bx + c = 0, the quadratic formula provides a general solution for the zeros: x = (-b ± √(b² - 4ac)) / (2a). This formula is a powerful tool that can be used to find the zeros of any quadratic equation, regardless of whether it can be easily factored. In our case, the quadratic factor is 3x² - 5x - 2, so a = 3, b = -5, and c = -2. Plugging these values into the quadratic formula, we get:
x = (5 ± √((-5)² - 4 * 3 * -2)) / (2 * 3) x = (5 ± √(25 + 24)) / 6 x = (5 ± √49) / 6 x = (5 ± 7) / 6
This gives us two solutions:
x = (5 + 7) / 6 = 12 / 6 = 2 x = (5 - 7) / 6 = -2 / 6 = -1/3
Therefore, the zeros of the quadratic factor 3x² - 5x - 2 are x = 2 and x = -1/3.
The Complete Set of Zeros: A Comprehensive Solution
Having found the zeros of both factors, we can now state the complete set of zeros for the polynomial function f(x) = 3x³ - 8x² + 3x + 2. We started with the given zero, x = 1, and found two additional zeros, x = 2 and x = -1/3. Thus, the zeros of f(x) are 1, 2, and -1/3.
Returning to the original question, we were asked to identify another zero of f(x) from a set of options. Based on our analysis, the other zero of f(x) is:
A. x = 2
This exploration has demonstrated the interconnectedness of concepts in algebra, from the Factor Theorem to synthetic division and the quadratic formula. By systematically applying these tools, we have successfully found all the zeros of the given polynomial function.
Finding Zeros of Polynomials: A Step-by-Step Guide to Solving f(x) = 3x³ - 8x² + 3x + 2
Polynomials are essential mathematical expressions used to model a wide array of real-world phenomena. The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. These zeros, also known as roots, are crucial in understanding the behavior of the polynomial function and have significant applications in various scientific and engineering disciplines. In this detailed guide, we will walk through the process of finding the zeros of the polynomial function f(x) = 3x³ - 8x² + 3x + 2, given that x = 1 is one of its zeros. We'll use a combination of algebraic techniques to identify the other zeros of this cubic polynomial.
The Power of the Remainder Theorem: Verifying the Given Zero
Before diving into finding the remaining zeros, it's good practice to verify the given zero. The Remainder Theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is f(a). In our case, we want to verify that x = 1 is a zero of f(x) = 3x³ - 8x² + 3x + 2. According to the Remainder Theorem, we need to evaluate f(1):
f(1) = 3(1)³ - 8(1)² + 3(1) + 2 f(1) = 3 - 8 + 3 + 2 f(1) = 0
Since f(1) = 0, we confirm that x = 1 is indeed a zero of the polynomial f(x). This verification step reinforces our starting point and builds confidence as we proceed to find the other zeros.
From Zero to Factor: Applying the Factor Theorem
As discussed earlier, the Factor Theorem provides a direct connection between the zeros of a polynomial and its factors. Since x = 1 is a zero of f(x), the Factor Theorem tells us that (x - 1) is a factor of f(x). This is a crucial piece of information that allows us to reduce the cubic polynomial into a lower-degree polynomial. To find the other factor, we can use polynomial division.
Polynomial Division: Uncovering the Remaining Factors
Polynomial division is a method for dividing a polynomial by another polynomial of lower or equal degree. In our case, we will divide f(x) = 3x³ - 8x² + 3x + 2 by (x - 1). We can use either polynomial long division or synthetic division. For this explanation, we'll continue with the synthetic division method, which we introduced earlier. Setting up the synthetic division:
1 | 3 -8 3 2
|
-----------
Performing the synthetic division, we get:
1 | 3 -8 3 2
| 3 -5 -2
-----------
3 -5 -2 0
The quotient is 3x² - 5x - 2, and the remainder is 0, confirming that (x - 1) is a factor. Therefore, we can write f(x) as:
f(x) = (x - 1)(3x² - 5x - 2)
Now, we need to find the zeros of the quadratic factor 3x² - 5x - 2.
Factoring the Quadratic: A Path to the Final Zeros
To find the zeros of the quadratic 3x² - 5x - 2, we can attempt to factor it. We are looking for two binomials that multiply to give 3x² - 5x - 2. We need two numbers that multiply to (3 * -2 = -6) and add up to -5. These numbers are -6 and 1. We can rewrite the quadratic as:
3x² - 5x - 2 = 3x² - 6x + x - 2
Now, we factor by grouping:
3x² - 6x + x - 2 = 3x(x - 2) + 1(x - 2) 3x² - 5x - 2 = (3x + 1)(x - 2)
So, the factored form of the quadratic is (3x + 1)(x - 2).
Identifying All Zeros: Completing the Solution
Now that we have factored the quadratic, we can write the complete factored form of f(x):
f(x) = (x - 1)(3x + 1)(x - 2)
To find the zeros, we set each factor equal to zero and solve for x:
x - 1 = 0 => x = 1 3x + 1 = 0 => x = -1/3 x - 2 = 0 => x = 2
Thus, the zeros of f(x) = 3x³ - 8x² + 3x + 2 are 1, -1/3, and 2.
Therefore, the other zero of f(x), as asked in the original question, is:
A. x = 2
In this comprehensive guide, we have demonstrated the step-by-step process of finding the zeros of a polynomial, using a combination of the Remainder Theorem, the Factor Theorem, polynomial division, and factoring quadratic equations. These techniques are fundamental in algebra and provide a powerful framework for solving a wide range of polynomial problems.
