Given the function;
[tex]g(x)=2(7x-2)^3(4-9x^2)[/tex]
To find the zeros of this function, we shall set the function equal to zero and solve for the variable x. This is shown below;
[tex]\begin{gathered} \text{We start by factorizing the right side of the function;} \\ 2(7x-2)^3(4-9x^2) \\ =2(7x-2)^3(-9x^2+4) \\ \text{Factor out -1 and you'll have;} \\ =2(7x-2)^3-1(9x^2-4) \\ =2(7x-2)^3-1(3x+2)(3x-2) \\ =-2(7x-2)^3(3x+2)(3x-2) \end{gathered}[/tex]
We now have;
[tex]-2(7x-2)^3(3x+2)(3x-2)=0[/tex]
Next we'll apply the zero factor principle which states that;
[tex]\begin{gathered} If \\ ab=0 \\ \text{Then,} \\ a=0,\text{ or b}=0 \end{gathered}[/tex]
We will now have the following;
[tex]\begin{gathered} 7x-2=0 \\ 7x=2 \\ x=\frac{2}{7} \end{gathered}[/tex][tex]\begin{gathered} 3x+2=0 \\ 3x=-2 \\ x=-\frac{2}{3} \end{gathered}[/tex][tex]\begin{gathered} 3x-2=0 \\ 3x=2 \\ x=\frac{2}{3} \end{gathered}[/tex]
To calculate the y-intercept, we simply find the value of the equation when x = 0.
We'll now have the following;
[tex]\begin{gathered} y=2(7x-2)^3(4-9x^2) \\ y=2(7\lbrack0\rbrack-2)^3(4-9\lbrack0\rbrack^2) \\ y=2(0-2)^3(4-0) \\ y=2(-2)^3(4) \\ y=2(-8)(4) \\ y=-64 \end{gathered}[/tex]
ANSWER:
[tex]\begin{gathered} \text{The zeros of the function are;} \\ x=\frac{2}{7},x=-\frac{2}{3},x=\frac{2}{3} \\ y-\text{intercept}=-64 \end{gathered}[/tex]
