Question

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

Answer 1

Given:

$f(x)=4\mleft(x+7\mright)\mleft(x-5\mright)^2$

To find the real zero and its multiplicity:

Let the function equals to zero we get,

$\begin{gathered} f(x)=0 \\ 4\mleft(x+7\mright)\mleft(x-5\mright)^2=0 \\ x+7=0 \\ \Rightarrow x=-7 \\ x-5=0 \\ \Rightarrow x=5 \end{gathered}$

Hence, the real zeros are -7 and 5.

The zero -7, multiplicity 1, crosses the x axis.

The zero 5, multiplicity 2, touches the x axis.

Hence, the correct option is the first one.