Question

graph the polynomial function f(x)=(x - 1)(x + 4)^2 using parts (a) through (e). (c) determine the zeros of the function and their multiplicity. use this information to determine whether the graph crosses or touches the x - axis at each x - intercept. the zero(s) of f is/are 1, - 4. (simplify your answer. type an integer or a fraction. use a comma to separate answers as needed. type each answer only once.) the lesser zero of the function is of multiplicity , so the graph of f the x - axis at x=. the greater zero of the function is of multiplicity , so the graph of f the x - axis at x=.

Explanation:

Step1: Identify zero - values and multiplicities

The function is $f(x)=(x - 1)(x + 4)^2$. The zero - values are found by setting $f(x)=0$. So, $x-1 = 0$ gives $x = 1$ and $(x + 4)^2=0$ gives $x=-4$. The factor $(x - 1)$ has multiplicity 1 and the factor $(x + 4)^2$ has multiplicity 2.

Step2: Determine graph - x - axis behavior

If a zero $c$ of a polynomial function $y = f(x)$ has odd multiplicity, the graph of the function crosses the $x$ - axis at $x = c$. If a zero $c$ has even multiplicity, the graph of the function touches the $x$ - axis at $x = c$.
The lesser zero is $x=-4$ with multiplicity 2 (even), so the graph of $f$ touches the $x$ - axis at $x=-4$. The greater zero is $x = 1$ with multiplicity 1 (odd), so the graph of $f$ crosses the $x$ - axis at $x = 1$.

Answer:

The lesser zero of the function is of multiplicity 2, so the graph of $f$ touches the $x$-axis at $x=-4$. The greater zero of the function is of multiplicity 1, so the graph of $f$ crosses the $x$-axis at $x = 1$.