Question

Given polynomial \[f\left( x \right) = {x^3} + 2{x^2} - 51x + 108\] and a factor x + 9, how do you find all other factors?

Answer 1

Hint: We will first divide the given polynomial f(x) by the given factor x + 9 and the quotient we thus obtain will be a quadratic equation, then we will solve it to find the required factors.

Complete step by step solution:
We are given that we are required to find the factors of the given polynomial \[f\left( x \right) = {x^3} + 2{x^2} - 51x + 108\] other than (x + 9).
Let us first take (x + 9) as the divisor and the polynomial \[f\left( x \right) = {x^3} + 2{x^2} - 51x + 108\] as the dividend.
So, we will obtain the following expression:-
$ \Rightarrow x + 9)\overline {{x^3} + 2{x^2} - 51x + 108} $
Now, we will first multiply the divisor by ${x^2}$, we will then obtain the following expression:-
                  ${x^2}$
\[ \Rightarrow x + 9)\overline {{x^3} + 2{x^2} - 51x + 108} \]
                 \[\underline {{x^3} + 9{x^2}} \]
                        \[ - 7{x^2} - 51x + 108\]
Now, we will multiply the divisor by – 7x, we will then obtain the following expression:-
                  ${x^2} - 7x$
\[ \Rightarrow x + 9)\overline {{x^3} + 2{x^2} - 51x + 108} \]
                 \[\underline {{x^3} + 9{x^2}} \]
                        \[ - 7{x^2} - 51x + 108\]
                         \[\underline { - 7{x^2} - 63x} \]
                          \[12x + 108\]
Now, we will multiply the divisor by 12, we will then obtain the following expression:-
                  ${x^2} - 7x + 12$
\[ \Rightarrow x + 9)\overline {{x^3} + 2{x^2} - 51x + 108} \]
                 \[\underline {{x^3} + 9{x^2}} \]
                        \[ - 7{x^2} - 51x + 108\]
                         \[\underline { - 7{x^2} - 63x} \]
                          \[12x + 108\]
                          \[\underline {12x + 108} \]
                              0
Therefore, we can write the given polynomial \[f\left( x \right) = {x^3} + 2{x^2} - 51x + 108\] as \[{x^3} + 2{x^2} - 51x + 108 = (x + 9)({x^2} - 7x + 12)\]
We can definitely write this using the splitting of middle term method like the following expression:-
\[ \Rightarrow {x^3} + 2{x^2} - 51x + 108 = (x + 9)({x^2} - 3x - 4x + 12)\]
Taking x common from first two terms in the quadratic in right and – 4 common from last two terms in the quadratic above, we will then obtain the following equation:-$ \Rightarrow {x^2} - 7x + 12 = 0$
\[ \Rightarrow {x^3} + 2{x^2} - 51x + 108 = (x + 9)\left\{ {x(x - 3) - 4(x - 3)} \right\}\]
We can now write it as:-
\[ \Rightarrow {x^3} + 2{x^2} - 51x + 108 = (x + 9)(x - 3)(x - 4)\]
Hence, the other factors are (x – 3) and (x – 4).

Note: The students must note that you may use alternate methods for solving the equations other than using the method of “splitting the middle term”.
Alternate Way: This way is known for the use of quadratic formulas for finding the roots.
The general quadratic equation is given by $a{x^2} + bx + c = 0$ and its roots are given by:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Comparing it to the given equation, we have a = 1, b = - 7 and c = 12.
So, the roots of the equation are:-
$ \Rightarrow x = \dfrac{{ - ( - 7) \pm \sqrt {{{( - 7)}^2} - 4(1)(12)} }}{{2(1)}}$
Simplifying the calculations in the above equation, we will then obtain the following equation:-
$ \Rightarrow x = \dfrac{{7 \pm \sqrt {49 - 48} }}{2}$
Simplifying the calculations further in the above equation, we will then obtain the following equation:-
$ \Rightarrow x = \dfrac{{7 \pm 1}}{2}$
Thus, the roots are 3 and 4 and hence, the factors are (x – 3) and (x – 4).