Finding Polynomial Zeros, Intercepts, And End Behavior
Hey math enthusiasts! Today, we're diving deep into the world of polynomials. We'll be working with the cubic polynomial P(x) = -5x³ + 48x² - 97x + 42. Don't worry, it might seem intimidating at first, but trust me, we'll break it down step by step and make it super understandable. We'll explore its zeroes, y-intercept, and end behavior – everything you need to know to truly understand this function! So, grab your pencils, and let's get started!
a. Listing Possible Rational Zeroes: The Rational Root Theorem
Alright, guys, first things first! We need to find the possible rational zeroes of our polynomial. This is where the Rational Root Theorem comes to the rescue. This theorem provides us with a handy way to narrow down the potential rational roots. It states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where 'p' is a factor of the constant term (the number without an 'x') and 'q' is a factor of the leading coefficient (the number in front of the highest power of 'x').
In our case, the constant term is 42, and the leading coefficient is -5. So, let's find the factors!
Factors of 42 (p): ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42 Factors of -5 (q): ±1, ±5
Now, we'll create all possible fractions p/q. This will give us a list of potential rational zeroes. Let's make that list:
Possible Rational Zeroes: ±1/1, ±2/1, ±3/1, ±6/1, ±7/1, ±14/1, ±21/1, ±42/1, ±1/5, ±2/5, ±3/5, ±6/5, ±7/5, ±14/5, ±21/5, ±42/5
Simplifying, we get:
Possible Rational Zeroes: ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42, ±1/5, ±2/5, ±3/5, ±6/5, ±7/5, ±14/5, ±21/5, ±42/5
That's quite a list, right? But hey, don't worry! This list gives us a starting point. We can now use synthetic division or direct substitution to test these potential zeroes and see which ones actually make P(x) equal to zero. Remember, finding the zeroes is crucial because it helps us understand where the graph of the polynomial crosses the x-axis. This process is like a treasure hunt, and we're armed with the map (the Rational Root Theorem) to find the hidden gems (the zeroes).
Keep in mind that not all of these possible rational zeroes will necessarily be actual zeroes. Some might be, and some might not. We have to test them to find out. This theorem is a powerful tool to narrow down our options, making the process of finding the zeroes much more manageable. So, let's move on to the next step, where we'll actually find those zeroes!
b. Finding the Zeroes of P(x): Uncovering the Roots
Alright, we've got our list of possible rational zeroes. Now, it's time to find the actual zeroes of our polynomial P(x) = -5x³ + 48x² - 97x + 42. There are a couple of ways we can do this: we can use synthetic division or simply substitute the values into the polynomial to see if we get zero. Let's use a combination of both for efficiency, guys!
First, let's start testing some of the simpler values from our list of possible rational zeroes. Let's try x = 1:
P(1) = -5(1)³ + 48(1)² - 97(1) + 42 = -5 + 48 - 97 + 42 = -12
Nope, not a zero. Let's try x = 2:
P(2) = -5(2)³ + 48(2)² - 97(2) + 42 = -40 + 192 - 194 + 42 = 0
Bingo! We found one! x = 2 is a zero of P(x). That means (x - 2) is a factor of the polynomial. Now we can use synthetic division to find the other factor.
Here's how synthetic division works with x = 2:
2 | -5 48 -97 42
| -10 76 -42
------------------
-5 38 -21 0
The result gives us the quadratic expression -5x² + 38x - 21. Now, we have effectively reduced the cubic equation to a quadratic one, which is much easier to solve. To find the remaining zeroes, we can either factor this quadratic expression or use the quadratic formula.
Let's try factoring. We are looking for two numbers that multiply to give us (-5)(-21) = 105 and add up to 38. After some thinking, we find that the numbers are 3 and 35. So we can rewrite the quadratic equation like this:
-5x² + 38x - 21 = -5x² + 35x + 3x - 21
Now, factor by grouping:
= -5x(x - 7) + 3(x - 7)
= (x - 7)(-5x + 3)
Setting each factor equal to zero, we get:
x - 7 = 0 => x = 7 -5x + 3 = 0 => x = 3/5
So, the zeroes of P(x) are x = 2, x = 7, and x = 3/5. These are the x-values where the graph of the polynomial crosses the x-axis. We did it, guys! We successfully found all the zeroes of our cubic polynomial. This is a big step in understanding the behavior of the polynomial and its graph.
c. Identifying the Y-Intercept: Where the Graph Crosses
Now, let's pinpoint the y-intercept of the polynomial P(x) = -5x³ + 48x² - 97x + 42. Remember, the y-intercept is the point where the graph of the function intersects the y-axis. At this point, the x-coordinate is always zero. This means we simply need to substitute x = 0 into our polynomial equation and solve for P(x).
So, let's do that:
P(0) = -5(0)³ + 48(0)² - 97(0) + 42 = 0 + 0 - 0 + 42 = 42
Therefore, the y-intercept is at the point (0, 42). This tells us that the graph of the polynomial crosses the y-axis at the y-value of 42. The y-intercept is always easy to find because it's simply the constant term in the polynomial when it is in standard form. In this case, 42 is our constant, and that gives us the y-intercept coordinates.
Knowing the y-intercept provides us with another important piece of information about the graph of our polynomial. Combined with the zeroes we found earlier, we are building a more comprehensive picture of the graph's behavior. We now know where the graph crosses both the x-axis and the y-axis. This information will be especially helpful when we want to sketch or analyze the graph more deeply.
d. Describing the End Behavior: The Long-Run Trend
Finally, let's explore the end behavior of the graph of our polynomial P(x) = -5x³ + 48x² - 97x + 42. End behavior describes what happens to the graph of a function as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). In other words, we're looking at what happens to the function's y-values as we move far to the right or far to the left on the x-axis.
For polynomials, the end behavior is determined by two key things: the degree of the polynomial (the highest power of x) and the sign of the leading coefficient (the number in front of the term with the highest power of x).
In our case, the degree of the polynomial is 3 (because of the x³ term), which is an odd number. The leading coefficient is -5, which is negative.
Here's how we determine the end behavior based on these two pieces of information:
- Odd Degree and Negative Leading Coefficient: As x approaches positive infinity (x → ∞), P(x) approaches negative infinity (P(x) → -∞). And as x approaches negative infinity (x → -∞), P(x) approaches positive infinity (P(x) → ∞).
In simpler terms:
- As we go further to the right on the x-axis, the graph of P(x) goes down.
- As we go further to the left on the x-axis, the graph of P(x) goes up.
This end behavior is a characteristic of all cubic polynomials with a negative leading coefficient. The graph will start high on the left, cross the x-axis in three places (since we found three zeroes), and then end low on the right.
Understanding the end behavior of a polynomial gives us a crucial understanding of the overall shape and direction of the graph. It helps us visualize how the graph behaves as it extends towards infinity in both directions. With this information, and our knowledge of the zeroes and the y-intercept, we can create a fairly accurate sketch of the polynomial's graph. We've now explored all the key aspects of this cubic polynomial, from finding its zeroes to understanding its long-run behavior. Great job, everyone!
We did it, guys! We've successfully analyzed the polynomial P(x) = -5x³ + 48x² - 97x + 42, exploring its zeroes, intercepts, and end behavior. We employed the Rational Root Theorem, synthetic division, and a bit of factoring to find the zeroes. We easily identified the y-intercept. And we determined the end behavior based on the degree and leading coefficient. Each step has given us valuable insight into understanding the function and its graphical representation. Keep practicing, and you'll become pros at these types of problems in no time! Keep up the great work! You've got this!
