We are given the function:

$f(x) = -3x(4x^3 - 1)(7x^2 + 1)(x + 1)$

We need to determine the end behavior of this polynomial function. The end behavior of a polynomial is primarily influenced by two things:

1. The degree of the polynomial.
2. The sign of the leading term (i.e., the coefficient of the highest power of $x$).

Step 1: Determining the Degree

Each factor contributes to the degree of the polynomial:

• $x$ contributes 1 (from $x^1$).
• $4x^3 - 1$ contributes 3 (since the highest power is $x^3$).
• $7x^2 + 1$ contributes 2 (since the highest power is $x^2$).
• $x + 1$ contributes 1 (from $x^1$).

Thus, the total degree of the polynomial is:

$1 + 3 + 2 + 1 = 7$

This is a degree 7 polynomial.

Step 2: Leading Coefficient

The leading term comes from multiplying the highest power of $x$ from each factor: $-3x \cdot 4x^3 \cdot 7x^2 \cdot x = -3 \cdot 4 \cdot 7 \cdot x^{1+3+2+1} = -84x^7$

The leading coefficient is $-84$, and the degree is odd (7).

Step 3: End Behavior Analysis

For polynomials, the end behavior depends on the degree and sign of the leading coefficient:

• Odd degree and negative leading coefficient:
  • As $x \to -\infty$ (left), $f(x) \to \infty$.
  • As $x \to \infty$ (right), $f(x) \to -\infty$.

Thus, the function increases to the left and decreases to the right.

Correct Answer:

$\text{increasing to the left and decreasing to the right}$

Do you want more details or have any other questions?

Here are 5 related questions for further exploration:

1. How does the degree of a polynomial affect its end behavior?
2. What is the impact of changing the sign of the leading coefficient on the graph of a polynomial?
3. Can you find the x-intercepts of the function $f(x)$?
4. How do you determine whether a polynomial has symmetry (even, odd, or neither)?
5. What is the behavior of the function near critical points (like where it equals 0)?

Tip: Always consider the degree and leading coefficient to predict the end behavior of polynomials.