Analyzing Zeros And End Behavior Of F(x) = X³ + 8x² + X - 42

Let's dive into the function f(x) = x³ + 8x² + x - 42. We'll explore how to determine if certain values are zeros of this function and how to describe its end behavior—all without relying on technology. This means we'll be using good old-fashioned algebra and reasoning!

Part A: Determining Zeros of f(x) = x³ + 8x² + x - 42

To determine if f(x) has zeros at -7, 2, and -3, we need to evaluate f(x) at each of these values. A zero of a function is a value of x that makes the function equal to zero. In other words, if f(a) = 0, then a is a zero of f(x). We'll use direct substitution to find out:

1. Checking if -7 is a zero:

We substitute x = -7 into the function:

f(-7) = (-7)³ + 8(-7)² + (-7) - 42

f(-7) = -343 + 8(49) - 7 - 42

f(-7) = -343 + 392 - 7 - 42

f(-7) = -343 + 392 - 49

f(-7) = 49 - 49

f(-7) = 0

Since f(-7) = 0, -7 is indeed a zero of f(x). This means that (x + 7) is a factor of the polynomial f(x). When we substitute -7 for x, we meticulously calculate each term: (-7)³ is -343, 8(-7)² is 8 * 49 = 392, and we subtract 7 and 42. By summing these values, we find that the result is 0, confirming that -7 is a zero of the polynomial. The process of direct substitution is fundamental in algebra, allowing us to verify potential roots of polynomials and understand their behavior.

2. Checking if 2 is a zero:

Next, we substitute x = 2 into the function:

f(2) = (2)³ + 8(2)² + (2) - 42

f(2) = 8 + 8(4) + 2 - 42

f(2) = 8 + 32 + 2 - 42

f(2) = 42 - 42

f(2) = 0

Since f(2) = 0, 2 is also a zero of f(x). This tells us that (x - 2) is another factor of f(x). Substituting 2 for x, we compute (2)³ as 8, 8(2)² as 8 * 4 = 32, and add 2 before subtracting 42. The sum totals 0, confirming that 2 is a zero. Recognizing that 2 is a zero implies that (x - 2) is a factor, which is crucial for polynomial factorization. This method of verifying zeros through substitution is a cornerstone of polynomial analysis.

3. Checking if -3 is a zero:

Now, let's substitute x = -3 into the function:

f(-3) = (-3)³ + 8(-3)² + (-3) - 42

f(-3) = -27 + 8(9) - 3 - 42

f(-3) = -27 + 72 - 3 - 42

f(-3) = -30 + 72 - 42

f(-3) = 42 - 42

f(-3) = 0

Since f(-3) = 0, -3 is indeed a zero of f(x). This means that (x + 3) is a factor of the polynomial. Substituting -3 into the function, we calculate (-3)³ as -27, 8(-3)² as 8 * 9 = 72, and subtract 3 and 42. The computation results in 0, which validates that -3 is a zero. Knowing that -3 is a zero implies that (x + 3) is a factor of f(x), further aiding in the polynomial's factorization. Thus, we can confidently state that -3 is a zero of the function.

Explanation

We have shown without the use of technology that -7, 2, and -3 are zeros of f(x) by substituting each value into the function and verifying that the result is zero. This method relies on the Factor Theorem, which states that if f(a) = 0, then (x - a) is a factor of f(x). Furthermore, since f(x) is a cubic polynomial (degree 3), it can have at most three zeros. We have found three distinct zeros, so we have found all the zeros of the polynomial.

Part B: Describing the End Behavior of f(x) = x³ + 8x² + x - 42

The end behavior of a function describes what happens to the function's values (f(x)) as x approaches positive infinity (∞) and negative infinity (-∞). To determine the end behavior without using technology, we focus on the leading term of the polynomial, which in this case is x³.

Understanding the Leading Term

The leading term x³ dictates the end behavior because, for very large values of x (either positive or negative), the higher powers of x dominate the other terms in the polynomial. The coefficient of the leading term also plays a crucial role. In our case, the coefficient is 1, which is positive.

End Behavior Analysis

1. As x approaches positive infinity (x → ∞):

   When x is a large positive number, x³ will also be a large positive number. Since the coefficient is positive, f(x) will approach positive infinity. We can write this as: as x → ∞, f(x) → ∞. In simple terms, as x gets larger and larger in the positive direction, the value of the function also gets larger and larger in the positive direction.

2. As x approaches negative infinity (x → -∞):

   When x is a large negative number, x³ will also be a large negative number because a negative number raised to an odd power remains negative. Since the coefficient is positive, f(x) will approach negative infinity. We can write this as: as x → -∞, f(x) → -∞. This means that as x becomes increasingly negative, the value of the function also decreases without bound in the negative direction.

Visualizing the End Behavior

We can visualize this end behavior by imagining the graph of f(x). On the right side of the graph (as x goes to ∞), the graph goes up (f(x) goes to ∞). On the left side of the graph (as x goes to -∞), the graph goes down (f(x) goes to -∞). This is characteristic of a cubic function with a positive leading coefficient.

Formal Description

In summary, the end behavior of f(x) = x³ + 8x² + x - 42 can be described as follows:

• As x approaches positive infinity, f(x) approaches positive infinity.
• As x approaches negative infinity, f(x) approaches negative infinity.

This means that the function rises to the right and falls to the left, a typical behavior for cubic functions with a positive leading coefficient. This behavior is critical in understanding the overall shape and characteristics of the function's graph.

In conclusion, we successfully determined the zeros of the function f(x) = x³ + 8x² + x - 42 and described its end behavior without the use of technology. By understanding the Factor Theorem and focusing on the leading term of the polynomial, we can analyze key properties of the function. For further exploration of polynomial functions and their properties, consider visiting a trusted educational resource like Khan Academy's Algebra Section.