Question

If ${{\rm{x}}^2} + {\rm{x}} - 12$ divides ${\rm{P}}\left( {\rm{x}} \right) = {{\rm{x}}^3} + {\rm{a}}{{\rm{x}}^3} + {\rm{bx}} - 84$ exactly, find the values of a and b.

Answer 1

Hint:
First factories ${{\rm{x}}^2} + {\rm{x}} - 12$ to get possible values of x. Then divide P(x) by ${{\rm{x}}^2} + {\rm{x}} - 12$ to form an equation. Putting in values of x, we get an equation in terms of a and b. Solve these equations for the values of a and b.

Complete step by step solution:
 ${\rm{g}}\left( {\rm{x}} \right) = {{\rm{x}}^2} + {\rm{x}} - 12 = 0$
 $ \Rightarrow {{\rm{x}}^2} + 4{\rm{x}} - 3{\rm{x}} - 12 = 0$
 $ \Rightarrow {\rm{x}}\left( {{\rm{x}} + 4} \right) - 3\left( {{\rm{x}} + 4} \right) = 0$
 $ \Rightarrow \left( {{\rm{x}} + 4} \right)\left( {{\rm{x}} - 3} \right) = 0$
 $ \Rightarrow {\rm{x}} = 3,{\rm{\;}} - 4$
 ${\rm{p}}\left( {\rm{x}} \right) = {{\rm{x}}^3} + {\rm{a}}{{\rm{x}}^2} + {\rm{bx}} - 84 = 0$
Since, g(x)completely divides p(x) remainder should be 0.
 ${\rm{p}}\left( {\rm{x}} \right) \div {\rm{g}}\left( {\rm{x}} \right) = {\rm{a}}{{\rm{x}}^2} - {{\rm{x}}^2} + {\rm{bx}} + 12{\rm{x}} - 84 = 0$
We get it by $\left( {{{\rm{x}}^3} + {\rm{a}}{{\rm{x}}^2} + {\rm{bx}} - 84} \right) \div {{\rm{x}}^2} + {\rm{x}} - 12 = 0$
Let x = 3
 Putting to equation (i), we get
 $27{\rm{a}} - 9 + 3{\rm{b}} + 36 - 84 = 0$
 $ \Rightarrow 9{\rm{a}} + 3{\rm{b}} = 57$
 $ \Rightarrow 3\left( {3{\rm{a}} + {\rm{b}}} \right) = 3 \times 19$
 $ \Rightarrow 3{\rm{a}} + {\rm{b}} = 19$ (ii)
Let x = 4,
 $16{\rm{a}} - 16 - 4{\rm{b}} - 48 - 84 = 0$
 $ \Rightarrow 16{\rm{a}} - 4{\rm{b}} = 148$
 $ \Rightarrow 4{\rm{a}} - {\rm{b}} = 37$ (iii)
Adding (ii) & (iii)
 7a = 56
 $ \Rightarrow {\rm{a}} = 8$
Then, $3{\rm{a}} + {\rm{b}} = 19$
$ \Rightarrow {\rm{b}} = 19 - 3\left( 8 \right) = 19 - 24 = - 5$
$ \Rightarrow {\rm{b}} = - 5$

So, a = 8, b = -5

Note:
It is an important step to find the values of x from the divisor. Students mostly start dividing the equations, then get confused in further steps.