Questions: Determine the end behavior, y-intercept, x-intercept(s) and multiplicities for the polynomial below, and then sketch a graph. f(x)=-3 x(x-1)^3(x+2)^2

Solution Steps

The polynomial function is \( f(x) = -3x(x-1)^3(x+2)^2 \). The degree of the polynomial is the sum of the exponents: \(1 + 3 + 2 = 6\). Since the leading term is negative, the end behavior is:

• As \( x \to \infty \), \( f(x) \to -\infty \).
• As \( x \to -\infty \), \( f(x) \to -\infty \).

The \( y \)-intercept is found by evaluating \( f(0) \): \[ f(0) = -3 \cdot 0 \cdot (0-1)^3 \cdot (0+2)^2 = 0 \] Thus, the \( y \)-intercept is \( (0, 0) \).

The \( x \)-intercepts occur where \( f(x) = 0 \): \[ -3x(x-1)^3(x+2)^2 = 0 \] The solutions are:

• \( x = 0 \) with multiplicity 1
• \( x = 1 \) with multiplicity 3
• \( x = -2 \) with multiplicity 2

Final Answer