Questions: Given the function P(x)=(x-1)^2(x-4), find the following: a) y-intercept as an ordered pair: b) x-intercepts as ordered pairs: c) When x → ∞, f(x) → ? d) When x → -∞, f(x) → ?

Solution Steps

a) To find the y-intercept of the function \( P(x) = (x-1)^2(x-4) \), evaluate the function at \( x = 0 \).

b) To find the x-intercepts, set the function equal to zero and solve for \( x \). This involves finding the roots of the equation \( (x-1)^2(x-4) = 0 \).

c) To determine the behavior of \( f(x) \) as \( x \rightarrow \infty \), analyze the leading term of the polynomial, which is \( x^3 \).

d) Similarly, to determine the behavior of \( f(x) \) as \( x \rightarrow -\infty \), analyze the leading term of the polynomial, which is \( x^3 \).

To find the y-intercept of the function \( P(x) = (x - 1)^2(x - 4) \), we evaluate the function at \( x = 0 \): \[ P(0) = (0 - 1)^2(0 - 4) = 1 \cdot (-4) = -4 \] Thus, the y-intercept is \( (0, -4) \).

To find the x-intercepts, we set the function equal to zero: \[ P(x) = (x - 1)^2(x - 4) = 0 \] This gives us the solutions: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \quad \text{(with multiplicity 2)} \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] Thus, the x-intercepts are \( (1, 0) \) and \( (4, 0) \).

As \( x \) approaches infinity, the leading term of the polynomial \( P(x) \) dominates. Since the leading term is \( x^3 \), we have: \[ \lim_{x \to \infty} P(x) = \infty \]

Similarly, as \( x \) approaches negative infinity, the leading term \( x^3 \) also dominates: \[ \lim_{x \to -\infty} P(x) = -\infty \]

Final Answer