Question

describe the key features of g(x), including theend behavior, y-intercept, and zeros.g(x) = x^3 - x^2- 4x + 4 than create a graph of the polynomial function

Answer 1

$g(x)=x^3-x^2-4x+4$

End behavior: You have the leading coefficient (number write in front of the variable with the largest exponent) of +1 and the polynomial degree is odd (3).

For this features the end behavior is: falls to left and rises to right.

Y-intercept: The value of g(x) when it cross the y-axis (when x is 0)

Substitute the x in the equation by 0 and evaluate the function:

$\begin{gathered} g(0)=0^3-0^2-4(0)+4 \\ g(0)=4 \end{gathered}$

y-intercept: 4

To find the zeros:

Equals the function to 0

$x^3-x^2-4x+4=0$

Factor:

-Factor x²

$x^2(x-1)-4x+4=0$

-Factor -4:

$x^2(x-1)-4(x-1)=0$

-Factor (x-1):

$(x-1)(x^2-4)=0$

Write the second factor as a substraction of squares:

$\begin{gathered} (x-1)(x^2-2^2)=0 \\ \\ a^2-b^2=(a+b)(a+b) \\ \\ (x-1)(x+2)(x-2)=0 \end{gathered}$

Equal each factor to 0 and solve x:

$\begin{gathered} x-1=0 \\ x=1 \\ \\ x+2=0 \\ x=-2 \\ \\ x-2=0 \\ x=2 \end{gathered}$

Then, the zeros of g(x) are: x=1, -2, 2