Given:
There are given the function:
[tex]f(x)=x^3+4x^2-9x-36[/tex]
Explanation:
To the factor, the above function, first find the first zero of the above function:
So,
From the function:
[tex]\begin{gathered} f(x)=x^{3}+4x^{2}-9x-36 \\ f\mleft(x\mright)=(x+4)(x^2-9) \end{gathered}[/tex]
Then,
[tex]\begin{gathered} f(x)=(x+4)(x^{2}-9) \\ f\mleft(x\mright)=(x+4)(x+3)(x-3) \end{gathered}[/tex]
So,
The factor of the given function is shown below:
[tex]f(x)=(x+4)(x+3)(x-3)[/tex]
Now,
Solve the given inequality:
[tex]x^3+4x^2-9x-36\leq0[/tex]
Then,
[tex]\begin{gathered} x^3+4x^2-9x-36\leq0 \\ (x+4)(x+3)(x-3)\leq0 \\ x\leq0\text{ or -3}\leq x\leq3 \end{gathered}[/tex]
Final answer:
Hence, the factor and the solution to the given inequality are shown below;
[tex]\begin{gathered} factor=(x+4)(x+3)(x-3) \\ Soution\text{ of inequality= x}\leq-4\text{ or -3}\leq x\leq3 \end{gathered}[/tex]
The number line graph of the inequality is shown below:
From the above graph, we can see that the first value of x is less than and equal to -4 and for the second value, the x has lies between -3 and 3.
