Question

How do you find the zeros of \[f(x)={{x}^{3}}+4{{x}^{2}}-25x-100\]?

Answer 1

Hint: The degree of an equation is the highest power to which the variable is raised. The degree decides whether the equation is linear, quadratic, cubic, etc. If the degree equals 3, then the equation is cubic. To find the roots/ zeros of a quadratic equation, we need to first find one of the roots by hit and trial method. Then write the equation as a product of the factor for the root and a quadratic equation. We can easily find the roots of the equation using various methods.

Complete step-by-step answer:
We are given \[f(x)={{x}^{3}}+4{{x}^{2}}-25x-100\], we need to find the roots of this function, which means we have to solve the equation \[{{x}^{3}}+4{{x}^{2}}-25x-100=0\].
As we can see that, if we substitute \[x=-4\] in the given function, we get
\[\begin{align}
  & \Rightarrow f(-4)={{\left( -4 \right)}^{3}}+4{{\left( -4 \right)}^{2}}-25\left( -4 \right)-100 \\
 & \Rightarrow f(-4)=-64+64+100-100=0 \\
\end{align}\]
Hence, \[x=-4\] is a root of the equation \[{{x}^{3}}+4{{x}^{2}}-25x-100=0\]. Hence, we can write the equation as,
\[\begin{align}
  & \Rightarrow {{x}^{3}}+4{{x}^{2}}-25x-100=\left( x-(-4) \right)\left( {{x}^{2}}-25 \right) \\
 & \Rightarrow \left( x+4 \right)\left( {{x}^{2}}-25 \right) \\
\end{align}\]
Thus, we get it as a product of a factor of a factor and a quadratic equation. Now, we need to find the roots of the equation \[{{x}^{2}}-25\]. As this expression is of the form \[{{a}^{2}}-{{b}^{2}}\], we can write it as,
 \[\Rightarrow {{x}^{2}}-25=\left( x+5 \right)\left( x-5 \right)\]
From the above factored form, the roots of the equation \[{{x}^{2}}-25\] are \[x=-5\] or \[x=5\]
So, we get the roots of the cubic equation \[{{x}^{3}}+4{{x}^{2}}-25x-100=0\] as, \[x=-4\] or \[x=-5\] or \[x=5\].

Note: To find the roots of a cubic equation, we need to find one root by the method of hit and trial. For this we need to check the value of the equation at \[0,\pm 1,\pm 2\]. In most of the cases, one of these values will be the root of the equation. If not, then we have to factorize the equation.