The given polynomial function is:
[tex]f(x)=-2(x-1)(x+3)^2[/tex]a. Find any real zeros of f.
To find the real zeros, we need to find the x-values that make each of the factors equal to zero, so, the factors of the polynomial function are:
[tex]\begin{gathered} -2 \\ (x-1) \\ (x+3)^2 \end{gathered}[/tex]Equal (x-1) and (x+3) to zero and solve for x:
[tex]\begin{gathered} x-1=0 \\ x=1\text{ Real zero} \\ x+3=0 \\ x=-3\text{ Real zero} \end{gathered}[/tex]The real zeros of f are 1,-3
The multiplicity is given by the power of the factor, then 1 has multiplicity 1 and -3 has multiplicity 2, so:
The multiplicit of the larger zero is 1
The multiplicity of the smaller zero is 2
b. Determine whether the graph crosses or touches the x-axis at each x-intercept.
When multiplicity is even the graph touches the x-axis and when is odd the graph crosses the x-axis.
Then, the graph crosses the x-axis at the larger x-intercept.
The graph touches the x-axis at the smaller x-intercept.
c. The maximum number of turning points is determined by the degree of the polynomial, we can rewrite the polynomial as:
[tex]\begin{gathered} f(x)=(-2x+2)(x+3)^2 \\ f(x)=(-2x+2)(x^2+2*3*x+3^2) \\ f(x)=(-2x+2)(x^2+6x+9) \\ f(x)=-2x*x^2-2x*6x-2x*9+2x^2+2*6x+2*9 \\ f(x)=-2x^3-12x^2-18x+2x^2+12x+18 \\ f(x)=-2x^3-10x^2-6x+18 \end{gathered}[/tex]The degree (n) is the greatest power, then this polynomial is degree n=3, and the number of turning points is given by n-1, so:
The maximum number of turning points on the graph is 2
d. Type the power function that the graph of f resembles for large values of |x|
We can apply the end behavior theorem which states: for larger values of |x| the graph of the polynomial:
[tex]f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0[/tex]resembles the graph of the power function:
[tex]y=a_nx^n[/tex]Thus, the power function that the graph of f resembles for larger values of |x| is:
[tex]y=-2x^3[/tex]