Question

If ax³+bx²-5x+2 has x+2as a factor and leaves a remainder 12 when divided by x-2 find the values of a and b.​

Answer 1

GIVEN:-

• ax³+bx²-5x+2 has x+2 as a factor and leaves a remainder 12 when divided by x-2.

TO FIND:-

• The Values of a and b.

Now,

$\implies\rm{x+2=0}$

$\implies\rm{x=-2}$

Now,put the value of "x" in p(x).

$\implies\rm{P(x)=ax^3+bx^2-5x+2}$

$\implies\rm{P(-2)=a(-2)^3+b(-2)^2-5(-2)+2}$

$\implies\rm{-8a+4b+10+2=0}$

$\implies\rm{-8a+4b+12=0}$

$\implies\rm{-2a+b+3=0}$

$\implies\rm{-2a+b=-3}$.............1

Again,

$\implies\rm{x-2=0}$

$\implies\rm{x=2}$

Now,put the value of "x" in p(x).

$\implies\rm{P(x)=ax^3+bx^2-5x+2}$

$\implies\rm{P(2)=a(2)^3+b(2)^2-5(2)+2}$

$\implies\rm{8a+4b-10+2=12}$

$\implies\rm{8a+4b-8=12}$

$\implies\rm{2a+b-2=3}$...........2

Adding equation.1 and equation 2.

$\implies\rm{-2a+b+2a+b-2=-3+3}$

$\implies\rm{2b=2}$

$\implies\rm{b=\dfrac{\cancel{2}}{\cancel{2}}}$

$\implies\rm{b=1}$

Now,Put the value of b in eq.2

$\implies\rm{2a+b-2=3}$

$\implies\rm{2a+1-4=3}$

$\implies\rm{2a-3=3}$

$\implies\rm{2a=6}$

$\implies\rm{a=\dfrac{6}{2}}$

$\implies\rm{a=3}$

Hence,The value of a and b is 3 and 1 respectively.