Question

Find the roots of f(x)=x^3 + 2x^2 +3x +4

Answer 1

f(x) will not have any roots.

Solution:

The roots of f(x) means the solution of f(x) = 0 which are the points where the function f(x) crosses the x axis. As the given equation, the highest power is 3, hence the equation will have total 3 roots.

Let us assume the three roots are a, b, c

Hence,
`x^(3)+2 x^(2)+3 x+4=(x-a)(x-b)(x-c)=0`

Multiplying the brackets we get
`a* b* c =4`

So a, b, c must be the factors of 4

The possibilities of factors of 4 are +1, -1, +2, -2, +4, -4

Substituting the values we get,

`f(1)=1^(3)+2 * 1^(2)+3 * 1+4=10`

`f(-1)=\left(-1^(3)\right)+2 *(-1)^(2)+3 *(-1)+4=-1+2-3+4=2`

`f(2)=2^(3)+2 * 2^(2)+3 * 2+4=26`

`f(-2)=\left(-2^(3)\right)+2 *(-2)^(2)+3 *(-2)+4=-2`

`f(4)=4^(3)+2 * 4^(2)+3 * 4+4=112`

`f(-4)=-4^(3)+2 *(-4)^(2)+3 *(-4)+4=-40`

So, there are no values that satisfy the equation.

Hence f(x) will not have any roots.