Hey everyone! Today, we're diving into a fascinating polynomial function, f(x) = 2x⁴ - x³ - 18x² + 9x. Our mission? To find its factors and pinpoint its zeros. This is a crucial skill in algebra and calculus, and it opens doors to understanding the behavior of functions. Let's break it down step by step, making it super clear and easy to follow.
Factoring the Polynomial f(x) = 2x⁴ - x³ - 18x² + 9x
First things first, let's talk about factoring. When we see a polynomial like this, our initial goal is to simplify it by breaking it down into smaller, more manageable pieces. This is where factoring comes in handy. By identifying common factors and using techniques like grouping, we can rewrite the polynomial in a way that reveals its structure more clearly. Factoring not only simplifies the expression but also paves the way for finding the zeros of the function. Zeros, in essence, are the x-values that make the function equal to zero, and they play a crucial role in understanding the graph and behavior of the polynomial.
When tackling a polynomial like f(x) = 2x⁴ - x³ - 18x² + 9x, the very first thing we should look for is a common factor. This is like the golden rule of factoring! In our case, we can see that each term has 'x' in it. So, let's factor out an 'x':
f(x) = x(2x³ - x² - 18x + 9)
Now, our expression looks a bit simpler, but we're not done yet. We have a cubic polynomial (2x³ - x² - 18x + 9) inside the parentheses. Factoring cubics can seem daunting, but a common technique we can use is factoring by grouping. This method involves pairing terms and factoring out common factors from each pair. It's like solving a puzzle where we strategically rearrange and simplify until we uncover the hidden structure of the polynomial.
Let’s group the terms in the cubic polynomial like this:
(2x³ - x²) + (-18x + 9)
From the first group, we can factor out an x²:
x²(2x - 1)
And from the second group, we can factor out a -9:
-9(2x - 1)
Notice something cool? Both groups now have a common factor of (2x - 1). This is exactly what we want! We can factor out this common binomial:
(2x - 1)(x² - 9)
We're getting closer! But hold on, we're not quite finished. The term (x² - 9) looks familiar, doesn't it? It's a difference of squares! This is a special pattern that we can factor further. Remember the formula: a² - b² = (a + b)(a - b).
Applying this to (x² - 9), where a = x and b = 3, we get:
(x + 3)(x - 3)
So, putting it all together, our fully factored form of the cubic polynomial is:
(2x - 1)(x + 3)(x - 3)
And remember that 'x' we factored out at the beginning? Let’s not forget about that! Our fully factored form of f(x) is:
f(x) = x(2x - 1)(x + 3)(x - 3)
Woohoo! We've successfully factored the polynomial. Factoring is a fundamental skill in algebra, and mastering it allows us to solve equations, simplify expressions, and gain insights into the behavior of functions. Now that we have the factored form, let's move on to finding the zeros of the function. The zeros are the values of x that make the function equal to zero, and they are closely related to the factors we just found. By setting each factor equal to zero and solving for x, we can uncover the zeros and paint a complete picture of the function's behavior.
Identifying the Zeros of f(x)
Now, let's use our factored form to find the zeros of the function. Remember, zeros are the x-values that make f(x) = 0. This is super important because the zeros tell us where the graph of the function crosses the x-axis. They are like key landmarks on the function's map, guiding us in understanding its behavior and shape. To find these zeros, we take each factor we found and set it equal to zero. It's like we're reversing the factoring process to pinpoint the values of x that make the whole expression vanish.
We have the factored form:
f(x) = x(2x - 1)(x + 3)(x - 3)
So, we set each factor to zero:
- x = 0
- 2x - 1 = 0
- x + 3 = 0
- x - 3 = 0
Let's solve each of these equations:
- x = 0 is already solved!
- For 2x - 1 = 0, add 1 to both sides and then divide by 2:
2x = 1
x = 1/2 3. For x + 3 = 0, subtract 3 from both sides:
x = -3 4. For x - 3 = 0, add 3 to both sides:
x = 3
So, we've found our zeros! The zeros of f(x) are x = 0, x = 1/2, x = -3, and x = 3. These values are incredibly significant because they tell us where the function's graph intersects the x-axis. Each zero corresponds to a point on the x-axis where the function's value is zero. This information is crucial for sketching the graph of the function and understanding its overall behavior. Knowing the zeros allows us to visualize the function's trajectory, identify its turning points, and gain insights into its increasing and decreasing intervals. It's like having a set of coordinates that help us navigate the function's landscape.
Completing the Statement: The Zeros of f(x)
Now, let's complete the statement using the zeros we've found. From left to right (meaning from the most negative to the most positive), the zeros are:
x = -3, x = 0, x = 1/2, and x = 3
So, our completed statement looks like this:
From left to right, function f has zeros at x = -3, x = 0, x = 1/2, and x = 3.
Awesome! We've successfully found the zeros and completed the statement. Finding the zeros of a function is a fundamental skill in algebra, and it has practical applications in various fields, including physics, engineering, and economics. The zeros provide valuable information about the function's behavior, such as its intersections with the x-axis, and they are essential for sketching the graph of the function.
Why This Matters: The Bigger Picture
You might be thinking,