Finding Zeros Of Polynomial Functions By Factoring A Comprehensive Guide

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Polynomial functions are fundamental in mathematics, and finding their zeros (the values of x for which the function equals zero) is a crucial skill. One powerful method for achieving this is factoring. In this article, we will guide you through the process of finding the zeros of the given function, $f(x) = 2x^4 - x^3 - 18x^2 + 9x$, by factoring. We'll break down each step, providing explanations and insights to enhance your understanding. By the end of this guide, you'll be equipped to tackle similar problems with confidence.

Step 1: Factoring out the Greatest Common Factor (GCF)

When dealing with polynomial functions, the first and most important step in factoring is to identify and factor out the Greatest Common Factor (GCF). The GCF is the largest expression that divides evenly into all the terms of the polynomial. This simplifies the polynomial and makes further factoring easier. In our case, the function is:

f(x)=2x4βˆ’x3βˆ’18x2+9xf(x) = 2x^4 - x^3 - 18x^2 + 9x

Observe that each term in the polynomial has a common factor of x. So, we can factor out x from the entire expression:

f(x)=x(2x3βˆ’x2βˆ’18x+9)f(x) = x(2x^3 - x^2 - 18x + 9)

By factoring out x, we have already found one zero of the function. Setting the factor x equal to zero gives us our first zero: x = 0. This is because when x is 0, the entire function becomes 0, satisfying the condition for a zero. The remaining expression inside the parentheses, $2x^3 - x^2 - 18x + 9$, is a cubic polynomial, which we will further factor in the next step. Factoring out the GCF not only simplifies the expression but also helps in identifying simple zeros, making the subsequent steps more manageable. Remember, always look for the GCF as the initial step in any factoring problem. This foundational step is crucial for solving polynomial equations and finding their roots efficiently.

Step 2: Factoring by Grouping

After factoring out the Greatest Common Factor (GCF), the next step in our journey to find the zeros of $f(x) = 2x^4 - x^3 - 18x^2 + 9x$ is to factor the cubic polynomial $2x^3 - x^2 - 18x + 9$. Factoring by grouping is a powerful technique used when dealing with polynomials that have four or more terms. This method involves grouping terms together and then factoring out common factors from each group. Let's apply this technique to our cubic polynomial. First, we group the first two terms and the last two terms:

(2x3βˆ’x2)+(βˆ’18x+9)(2x^3 - x^2) + (-18x + 9)

Now, we factor out the GCF from each group. From the first group, $2x^3 - x^2$, the GCF is $x^2$. Factoring this out, we get:

x2(2xβˆ’1)x^2(2x - 1)

From the second group, $-18x + 9$, the GCF is -9. Factoring this out, we get:

βˆ’9(2xβˆ’1)-9(2x - 1)

Notice that both groups now have a common binomial factor of $(2x - 1)$. This is a key indicator that factoring by grouping is working correctly. Now, we factor out the common binomial factor $(2x - 1)$ from the entire expression:

(2xβˆ’1)(x2βˆ’9)(2x - 1)(x^2 - 9)

At this point, we have successfully factored the cubic polynomial into a product of a binomial and a quadratic expression. This simplifies our task of finding the zeros, as we can now focus on factoring the quadratic expression further. Factoring by grouping is an invaluable tool in polynomial factoring, particularly when dealing with expressions that don't fit the standard quadratic factoring patterns. It allows us to break down complex polynomials into simpler, more manageable factors, making it easier to find the zeros of the function.

Step 3: Factoring the Difference of Squares

Having successfully factored the cubic polynomial by grouping, we've arrived at the expression $(2x - 1)(x^2 - 9)$. The next step in our quest to find the zeros of $f(x) = 2x^4 - x^3 - 18x^2 + 9x$ is to further factor the quadratic expression $x^2 - 9$. This quadratic expression is a classic example of the difference of squares, a pattern that appears frequently in algebra. The difference of squares pattern states that $a^2 - b^2$ can be factored as $(a + b)(a - b)$. In our case, $x^2 - 9$ can be seen as $x^2 - 3^2$, where a is x and b is 3. Applying the difference of squares pattern, we can factor $x^2 - 9$ as:

(x+3)(xβˆ’3)(x + 3)(x - 3)

Now, substituting this back into our expression, we have:

(2xβˆ’1)(x+3)(xβˆ’3)(2x - 1)(x + 3)(x - 3)

This is the completely factored form of the cubic polynomial. Combining this with the factor x we extracted in the first step, we have the fully factored form of the original function $f(x)$:

f(x)=x(2xβˆ’1)(x+3)(xβˆ’3)f(x) = x(2x - 1)(x + 3)(x - 3)

Factoring the difference of squares is a fundamental technique that significantly simplifies polynomial expressions. Recognizing and applying this pattern allows us to break down complex quadratics into linear factors, making it straightforward to find the zeros of the function. The ability to identify the difference of squares is a crucial skill in factoring and solving polynomial equations. With the function now fully factored, we are just one step away from finding all the zeros.

Step 4: Finding the Zeros

With the function $f(x) = 2x^4 - x^3 - 18x^2 + 9x$ fully factored as $f(x) = x(2x - 1)(x + 3)(x - 3)$, the final step in our journey is to find the zeros of the function. The zeros of a function are the values of x for which the function equals zero. In other words, they are the solutions to the equation $f(x) = 0$. Since we have factored the function, we can find the zeros by setting each factor equal to zero and solving for x. This is based on the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero.

So, we set each factor in $f(x) = x(2x - 1)(x + 3)(x - 3)$ equal to zero:

  1. x = 0

  2. 2xβˆ’1=02x - 1 = 0

  3. x+3=0x + 3 = 0

  4. xβˆ’3=0x - 3 = 0

The first equation, x = 0, already gives us one zero. For the second equation, $2x - 1 = 0$, we solve for x:

2x=12x = 1

x = rac{1}{2}

For the third equation, $x + 3 = 0$, we solve for x:

x=βˆ’3x = -3

And for the fourth equation, $x - 3 = 0$, we solve for x:

x=3x = 3

Thus, we have found four zeros for the function $f(x)$: 0, $ rac{1}{2}$, -3, and 3. These are the values of x where the function intersects the x-axis. Finding the zeros is a crucial step in understanding the behavior of polynomial functions. It allows us to sketch the graph of the function, determine its intervals of increase and decrease, and solve various mathematical problems. With all the zeros identified, we have successfully completed our analysis of the function $f(x)$.

Conclusion

In this article, we have walked through the process of finding the zeros of the polynomial function $f(x) = 2x^4 - x^3 - 18x^2 + 9x$ by factoring. We started by factoring out the Greatest Common Factor (GCF), then used factoring by grouping, followed by recognizing and applying the difference of squares pattern. Finally, we set each factor equal to zero to find the zeros of the function. The zeros we found are x = -3, x = 0, x = 1/2, and x = 3. This step-by-step approach not only helps in solving this particular problem but also provides a framework for tackling similar polynomial factoring problems. The ability to factor polynomials and find their zeros is a fundamental skill in algebra and calculus, with applications in various fields of mathematics, science, and engineering. By mastering these techniques, you can gain a deeper understanding of polynomial functions and their behavior.

Therefore, from left to right, the function $f$ has zeros at $x =$ -3, $x =$ 0, $x =$ 1/2, and $x =$ 3.