Questions: Given the function P(x)=x(x-7)(x+4), find its y-intercept is its x-intercepts are x1= , x2= and x3= with x1 ≤ x2 ≤ x3 When x → ∞, y → ∞ (Input + or - for the answer) When x → -∞, y → ∞ (Input + or - for the answer)

Solution Steps

To solve the given problem, we need to find the y-intercept, x-intercepts, and the end behavior of the polynomial function \( P(x) = x(x-7)(x+4) \).

1. Y-intercept: The y-intercept of a function is the value of the function when \( x = 0 \). Substitute \( x = 0 \) into the function to find the y-intercept.

2. X-intercepts: The x-intercepts are the values of \( x \) for which \( P(x) = 0 \). Set the function equal to zero and solve for \( x \).

3. End behavior: Analyze the leading term of the polynomial to determine the behavior of the function as \( x \) approaches positive and negative infinity. Since the leading term is \( x^3 \), the end behavior will be determined by the sign of the coefficient of \( x^3 \).

The y-intercept of a function is the value of the function when \( x = 0 \). For the polynomial \( P(x) = x(x-7)(x+4) \), substituting \( x = 0 \) gives: \[ P(0) = 0(0-7)(0+4) = 0 \] Thus, the y-intercept is \( \boxed{0} \).

The x-intercepts are the values of \( x \) for which \( P(x) = 0 \). Setting the polynomial equal to zero: \[ x(x-7)(x+4) = 0 \] The solutions are \( x = 0 \), \( x = 7 \), and \( x = -4 \). Therefore, the x-intercepts are \( x_1 = -4 \), \( x_2 = 0 \), and \( x_3 = 7 \). In sorted order, the x-intercepts are \( \boxed{-4} \), \( \boxed{0} \), and \( \boxed{7} \).

The end behavior of a polynomial is determined by its leading term. The leading term of \( P(x) = x(x-7)(x+4) \) is \( x^3 \). Since the coefficient of \( x^3 \) is positive, as \( x \rightarrow \infty \), \( y \rightarrow +\infty \), and as \( x \rightarrow -\infty \), \( y \rightarrow -\infty \).

Thus, the end behavior is:

• As \( x \rightarrow \infty \), \( y \rightarrow \boxed{+\infty} \).
• As \( x \rightarrow -\infty \), \( y \rightarrow \boxed{-\infty} \).

Final Answer