Answer:
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, [tex]x^{3}+2 x^{2}+3 x+4=(x-a)(x-b)(x-c)=0[/tex]
Multiplying the brackets we get [tex]a\times b\times c =4[/tex]
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,
[tex]f(1)=1^{3}+2 \times 1^{2}+3 \times 1+4=10[/tex]
[tex]f(-1)=\left(-1^{3}\right)+2 \times(-1)^{2}+3 \times(-1)+4=-1+2-3+4=2[/tex]
[tex]f(2)=2^{3}+2 \times 2^{2}+3 \times 2+4=26[/tex]
[tex]f(-2)=\left(-2^{3}\right)+2 \times(-2)^{2}+3 \times(-2)+4=-2[/tex]
[tex]f(4)=4^{3}+2 \times 4^{2}+3 \times 4+4=112[/tex]
[tex]f(-4)=-4^{3}+2 \times(-4)^{2}+3 \times(-4)+4=-40[/tex]
So, there are no values that satisfy the equation.
Hence f(x) will not have any roots.
