Question

what is the x-intercepts, y-intercepts, degree, sign of leading coefficient, left end behavior and right end behavior for h(x)=(x+1)(x-4)(x+3)

Answer 1

To analyze the polynomial function \( h(x) = (x+1)(x-4)(x+3) \), we can determine its x-intercepts, y-intercepts, degree, sign of the leading coefficient, and end behavior.

### 1. X-Intercepts
The x-intercepts occur when \( h(x) = 0 \). This happens when any of the factors is equal to zero.

- \( x + 1 = 0 \) → \( x = -1 \)
- \( x - 4 = 0 \) → \( x = 4 \)
- \( x + 3 = 0 \) → \( x = -3 \)

Thus, the x-intercepts are:
- \( (-1, 0) \)
- \( (4, 0) \)
- \( (-3, 0) \)

### 2. Y-Intercept
The y-intercept occurs when \( x = 0 \):

\[
h(0) = (0 + 1)(0 - 4)(0 + 3) = (1)(-4)(3) = -12
\]

So, the y-intercept is \( (0, -12) \).

### 3. Degree
The degree of a polynomial is the highest power of \( x \) in its expression. Since \( h(x) = (x + 1)(x - 4)(x + 3) \) is a product of 3 linear factors, the degree is:

\[
\text{Degree} = 3
\]

### 4. Sign of Leading Coefficient
To find the leading coefficient, we can expand the polynomial:

1. First, multiply the first two factors:
\[
(x + 1)(x - 4) = x^2 - 4x + x - 4 = x^2 - 3x - 4
\]

2. Next, multiply the result by the remaining factor:
\[
h(x) = (x^2 - 3x - 4)(x + 3)
\]
\[
= x^3 + 3x^2 - 3x^2 - 9x - 4x - 12 = x^3 - 13x - 12
\]

The leading term of the expanded polynomial \( x^3 \) indicates that the leading coefficient is \( 1 \).

### 5. Left End Behavior
For polynomials, the left end behavior is determined by the leading term. Since the leading term is positive and of odd degree, as \( x \to -\infty \), \( h(x) \to -\infty \).

### 6. Right End Behavior
Similarly, since the leading term is positive and of odd degree, as \( x \to +\infty \), \( h(x) \to +\infty \).

### Summary
- **X-intercepts:** \( (-1, 0) \), \( (4, 0) \), \( (-3, 0) \)
- **Y-intercept:** \( (0, -12) \)
- **Degree:** \( 3 \)
- **Leading Coefficient Sign:** Positive
- **Left End Behavior:** \( h(x) \to -\infty \) as \( x \to -\infty \)
- **Right End Behavior:** \( h(x) \to +\infty \) as \( x \to +\infty \)