Question

How do you find the factors of $f(x) = {x^3} + 2{x^2} - 23x - 60$?

Answer 1

Hint: We will find the first root of the given equation by hit and trial method and then, we will just divide using that factor and thus we find the other factors as well.

Complete step-by-step solution:
We are given that we need to find the factors of $f(x) = {x^3} + 2{x^2} - 23x - 60$.
Let us first try to put x = 1, we will obtain:
$ \Rightarrow $f (1) = 1 + 2 – 23 – 60 = - 80
Now, we can clearly see that it is way too less than 0. Let us try to put x = - 3.
$ \Rightarrow $f (-3) = -27 + 18 + 69 – 60 = 0
Hence, - 3 is a root of the given equation $f(x) = {x^3} + 2{x^2} - 23x - 60$.
Thus, (x + 3) is a factor of the given function $f(x) = {x^3} + 2{x^2} - 23x - 60$.
Let us divide $f(x) = {x^3} + 2{x^2} - 23x - 60$ by (x + 3).
$ \Rightarrow x + 3\mathop{\left){\vphantom{1{{x^3} + 2{x^2} - 23x - 60}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{{x^3} + 2{x^2} - 23x - 60}}}}
\limits^{\displaystyle \,\,\, {}}$
We will first multiply the divisor by ${x^2}$ and get the following expression:-
$ \Rightarrow x + 3\mathop{\left){\vphantom{1\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {{x^2}}}$
We will now multiply the divisor by $ - x$ and get the following expression:-
$ \Rightarrow x + 3\mathop{\left){\vphantom{1\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
  \underline { - {x^2} - 3x} \\
   - 20x - 60 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
  \underline { - {x^2} - 3x} \\
   - 20x - 60 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {{x^2} - x}}$
We will now multiply the divisor by – 20 and get the following expression:-
$ \Rightarrow x + 3\mathop{\left){\vphantom{1\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
  \underline { - {x^2} - 3x} \\
   - 20x - 60 \\
  \underline { - 20x - 60} \\
  0 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  {x^3} + 2{x^2} - 23x - 60 \\
  \underline {{x^3} + 3{x^2}} \\
   - {x^2} - 23x - 60 \\
  \underline { - {x^2} - 3x} \\
   - 20x - 60 \\
  \underline { - 20x - 60} \\
  0 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {{x^2} - x - 20}}$
Now, we can write the given expression as:-
$ \Rightarrow f(x) = {x^3} + 2{x^2} - 23x - 60 = (x + 3)({x^2} - x - 20)$
Now, we will factorize the other found quadratic equation.
Now, we will write ${x^2} - x - 20$ as ${x^2} + 4x - 5x - 20$.
Taking x common from first two and – 5 from next two terms in above equation:-
$ \Rightarrow {x^2} - x - 20 = x(x + 4) - 5(x + 4)$
Now, we will take (x + 4) common to get the following:-
$ \Rightarrow {x^2} - x - 20 = (x + 4)(x - 5)$
Thus, we have got:-
$ \Rightarrow f(x) = {x^3} + 2{x^2} - 23x - 60 = (x + 3)(x + 4)(x - 5)$

Hence, the factors of f (x) are (x + 3), (x + 4) and (x – 5).

Note: The students must note that they may find any other root instead of – 3 in the starting hit and trial method. Now, if they find any other root, they will get some other factor and then use it as a divisor to divide the given equation and thus get the other factors.
The students must note that you may use a quadratic formula to find the roots of the quadratic equation we obtained in it. We will still get the same answer.