Questions: Consider the following polynomial functions. f(x)=(x+1)^2(x-3) h(x)=x^4-3x^3 Choose the graph of each function from the choices below.

Transcript text: Consider the following polynomial functions. $f(x)=(x+1)^{2}(x-3)$ $h(x)=x^{4}-3 x^{3}$ Choose the graph of each function from the choices below.

Solution

Solution Steps

Step 1: Analyze f(x)

f(x) = (x+1)²(x-3) has zeros at x = -1 (multiplicity 2, so it touches the x-axis) and x = 3 (multiplicity 1, so it crosses the x-axis). The leading term is x³ indicating the graph rises to the right and falls to the left.

Step 2: Identify the graph of f(x)

Graph A matches the analysis: touches x-axis at x=-1, crosses at x=3, and has the correct end behavior.

Step 3: Analyze h(x)

h(x) = x⁴ - 3x³ = x³(x-3) has zeros at x = 0 (multiplicity 3, so it crosses the x-axis and has an inflection point) and x = 3 (multiplicity 1, so it crosses the x-axis). The leading term is x⁴, meaning both ends go in the same direction, upwards.

Step 4: Identify the graph of h(x)

Graph D matches the analysis: crosses at x=0 and x=3, has an inflection point at x=0 and has the correct end behavior.