Finding Zeros Of Polynomials A Detailed Guide To P(x)=(x^2+4x+3)(x^2-4)
Finding the zeros of a polynomial is a fundamental problem in algebra and has applications in various fields such as engineering, physics, and computer science. In this article, we will delve into the process of finding the zeros of the polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4). This polynomial is a product of two quadratic expressions, making it a fourth-degree polynomial. Our goal is to identify the values of x for which p(x) equals zero. This involves factoring the polynomial, setting each factor equal to zero, and solving for x. Let's break down the steps to effectively determine the zeros of this polynomial.
Factoring the Polynomial
To begin, we need to factor the given polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4). Factoring is the process of expressing a polynomial as a product of simpler polynomials. This makes it easier to find the zeros, as we can set each factor equal to zero and solve for x. We'll start by factoring each quadratic expression separately.
Factoring the First Quadratic: x² + 4x + 3
The first quadratic expression is x² + 4x + 3. To factor this, we look for two numbers that multiply to the constant term (3) and add up to the coefficient of the linear term (4). The numbers 3 and 1 satisfy these conditions since 3 * 1 = 3 and 3 + 1 = 4. Therefore, we can factor the quadratic as:
x² + 4x + 3 = (x + 3)(x + 1)
This factorization means that the quadratic expression equals zero when x = -3 or x = -1. These are two of the zeros of the polynomial p(x).
Factoring the Second Quadratic: x² - 4
The second quadratic expression is x² - 4. This is a difference of squares, which can be factored using the formula a² - b² = (a + b)(a - b). In this case, a = x and b = 2, so we have:
x² - 4 = (x + 2)(x - 2)
This factorization tells us that the quadratic expression equals zero when x = -2 or x = 2. These are the remaining two zeros of the polynomial p(x).
Complete Factorization of p(x)
Now that we've factored both quadratic expressions, we can write the complete factorization of the polynomial p(x) as:
p(x) = (x + 3)(x + 1)(x + 2)(x - 2)
This fully factored form allows us to easily identify all the zeros of the polynomial. The zeros are the values of x that make each factor equal to zero.
Identifying the Zeros
Now that we have the polynomial factored as p(x) = (x + 3)(x + 1)(x + 2)(x - 2), we can find the zeros by setting each factor equal to zero and solving for x. The zeros of a polynomial are the values of x that make the polynomial equal to zero. These values are also known as the roots of the polynomial equation.
Setting Each Factor to Zero
We have four factors in the factored form of p(x). To find the zeros, we set each factor equal to zero:
- x + 3 = 0
- x + 1 = 0
- x + 2 = 0
- x - 2 = 0
Solving for x
Now, we solve each equation for x:
- x + 3 = 0 Subtract 3 from both sides: x = -3
- x + 1 = 0 Subtract 1 from both sides: x = -1
- x + 2 = 0 Subtract 2 from both sides: x = -2
- x - 2 = 0 Add 2 to both sides: x = 2
The Zeros of the Polynomial
Therefore, the zeros of the polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4) are x = -3, x = -1, x = -2, and x = 2. These are the four values of x that make the polynomial equal to zero.
Verifying the Zeros
To ensure the accuracy of our results, it's essential to verify the zeros we found. We can do this by substituting each zero back into the original polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4) and confirming that the result is indeed zero. This process helps to catch any potential errors in our factorization or solving steps.
Substituting x = -3
Let's start by substituting x = -3 into the polynomial:
p(-3) = ((-3)² + 4(-3) + 3)((-3)² - 4)
p(-3) = (9 - 12 + 3)(9 - 4)
p(-3) = (0)(5)
p(-3) = 0
Since p(-3) = 0, x = -3 is indeed a zero of the polynomial.
Substituting x = -1
Next, we substitute x = -1 into the polynomial:
p(-1) = ((-1)² + 4(-1) + 3)((-1)² - 4)
p(-1) = (1 - 4 + 3)(1 - 4)
p(-1) = (0)(-3)
p(-1) = 0
Since p(-1) = 0, x = -1 is also a zero of the polynomial.
Substituting x = -2
Now, let's substitute x = -2 into the polynomial:
p(-2) = ((-2)² + 4(-2) + 3)((-2)² - 4)
p(-2) = (4 - 8 + 3)(4 - 4)
p(-2) = (-1)(0)
p(-2) = 0
Since p(-2) = 0, x = -2 is a zero of the polynomial.
Substituting x = 2
Finally, we substitute x = 2 into the polynomial:
p(2) = ((2)² + 4(2) + 3)((2)² - 4)
p(2) = (4 + 8 + 3)(4 - 4)
p(2) = (15)(0)
p(2) = 0
Since p(2) = 0, x = 2 is a zero of the polynomial.
Conclusion of Verification
We have successfully verified that x = -3, x = -1, x = -2, and x = 2 are indeed the zeros of the polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4). This verification step confirms the accuracy of our factorization and solving process.
Graphical Interpretation of Zeros
The zeros of a polynomial have a significant graphical interpretation. They represent the points where the graph of the polynomial intersects the x-axis. Each zero corresponds to an x-intercept on the graph. Understanding this connection between zeros and the graph can provide valuable insights into the behavior of the polynomial function. For our polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4), we found the zeros to be x = -3, x = -1, x = -2, and x = 2. This means that the graph of p(x) will cross the x-axis at these four points.
Visualizing the Graph
Imagine plotting the graph of p(x) on a coordinate plane. The graph would be a curve that crosses the x-axis at x = -3, x = -2, x = -1, and x = 2. These points are the x-intercepts and visually represent the zeros of the polynomial. The shape of the curve between these intercepts depends on the polynomial's other properties, such as its degree and leading coefficient.
Significance of X-Intercepts
The x-intercepts, or zeros, are crucial for understanding the behavior of the polynomial function. They tell us where the function's value is zero, which can be important in various applications. For instance, in physics, the zeros might represent equilibrium points in a system. In engineering, they could indicate critical values in a design.
Sketching the Graph
Knowing the zeros can help us sketch a rough graph of the polynomial. Since p(x) is a fourth-degree polynomial (quartic), it will have a general "W" or "M" shape, depending on the sign of the leading coefficient. In this case, the leading coefficient is positive (since the leading terms of the quadratics are x²), so the graph will have a "W" shape. The graph will start from the top left, cross the x-axis at x = -3, turn around, cross again at x = -2, turn again, cross at x = -1, turn again, cross at x = 2, and then continue upwards to the right.
Connection to Real-World Applications
The graphical interpretation of zeros is not just a theoretical concept; it has practical applications. For example, if p(x) represents the profit of a company as a function of the number of units sold (x), the zeros would represent the break-even points, where the company makes neither profit nor loss. Understanding these points is crucial for business decision-making.
Conclusion on Graphical Interpretation
In conclusion, the zeros of a polynomial are not just numerical solutions; they have a visual representation as the x-intercepts on the graph of the polynomial. This graphical interpretation provides a deeper understanding of the polynomial's behavior and its applications in various fields. For the polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4), the graph crosses the x-axis at x = -3, x = -1, x = -2, and x = 2, visually confirming our algebraic findings.
Conclusion
In this article, we've comprehensively explored the process of finding the zeros of the polynomial p(x) = (x^2 + 4x + 3)(x^2 - 4). We began by factoring the polynomial into its simplest forms, which allowed us to easily identify the values of x that make the polynomial equal to zero. Factoring involved breaking down the quadratic expressions into products of linear factors, a crucial step in solving polynomial equations. We then systematically set each factor to zero and solved for x, revealing the zeros of the polynomial: x = -3, x = -1, x = -2, and x = 2.
Verification and Accuracy
To ensure the accuracy of our findings, we rigorously verified each zero by substituting it back into the original polynomial. This step confirmed that each value indeed resulted in p(x) = 0, reinforcing the correctness of our solution. Verification is a vital practice in mathematics, as it helps to catch any potential errors in the algebraic manipulation and ensures the reliability of the results.
Graphical Significance
Furthermore, we discussed the graphical interpretation of the zeros. The zeros of a polynomial are the x-intercepts of its graph, representing the points where the graph crosses the x-axis. Understanding this connection provides a visual representation of the zeros and aids in sketching the graph of the polynomial. The graph of p(x), being a quartic polynomial with a positive leading coefficient, will have a “W” shape and intersect the x-axis at the zeros we found.
Practical Applications
Finding the zeros of polynomials is not just an academic exercise; it has practical applications in various fields. Zeros can represent critical points in physical systems, break-even points in business models, or solutions to engineering problems. The ability to find these zeros is a fundamental skill in mathematics and its applications.
Summary of the Process
In summary, finding the zeros of a polynomial involves a combination of algebraic techniques and conceptual understanding. Factoring the polynomial, setting each factor to zero, solving for the variable, verifying the solutions, and interpreting the zeros graphically are all essential steps in this process. By mastering these techniques, one can effectively solve a wide range of polynomial equations and gain insights into the behavior of polynomial functions.
Final Thoughts
The process of finding the zeros of a polynomial like p(x) = (x^2 + 4x + 3)(x^2 - 4) exemplifies the power and elegance of algebra. It demonstrates how complex expressions can be broken down into simpler components, allowing us to solve for unknown values and gain a deeper understanding of mathematical functions. This skill is invaluable not only in mathematics but also in numerous real-world applications, making it a cornerstone of mathematical education.
