Respuesta :
[tex]g(x)=x^3-x^2-4x+4[/tex]
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:
[tex]\begin{gathered} g(0)=0^3-0^2-4(0)+4 \\ g(0)=4 \end{gathered}[/tex]y-intercept: 4
To find the zeros:
Equals the function to 0
[tex]x^3-x^2-4x+4=0[/tex]Factor:
-Factor x²
[tex]x^2(x-1)-4x+4=0[/tex]-Factor -4:
[tex]x^2(x-1)-4(x-1)=0[/tex]-Factor (x-1):
[tex](x-1)(x^2-4)=0[/tex]Write the second factor as a substraction of squares:
[tex]\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}[/tex]Equal each factor to 0 and solve x:
[tex]\begin{gathered} x-1=0 \\ x=1 \\ \\ x+2=0 \\ x=-2 \\ \\ x-2=0 \\ x=2 \end{gathered}[/tex]