Question

Please can someone help me draw a graph for this question

Answer 1

Given:

There are given the function:

`f(x)=x^3+4x^2-9x-36`

Step-by-step explanation:

To the factor, the above function, first find the first zero of the above function:

So,

From the function:

`\begin{gathered} f(x)=x^(3)+4x^(2)-9x-36 \\ f\mleft(x\mright)=(x+4)(x^2-9) \end{gathered}`

Then,

`\begin{gathered} f(x)=(x+4)(x^(2)-9) \\ f\mleft(x\mright)=(x+4)(x+3)(x-3) \end{gathered}`

So,

The factor of the given function is shown below:

`f(x)=(x+4)(x+3)(x-3)`

Now,

Solve the given inequality:

`x^3+4x^2-9x-36\leq0`

Then,

`\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}`

Final answer:

Hence, the factor and the solution to the given inequality are shown below;

`\begin{gathered} factor=(x+4)(x+3)(x-3) \\ Soution\text{ of inequality= x}\leq-4\text{ or -3}\leq x\leq3 \end{gathered}`

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.