Question

The polynomial f (x) = ax^3 + x^2 - bx + 4 leaves
a remainder 10 when divided by (x - 2) while
the polynomial g (x) = ax^3 - 3x^2 + bx - 8 leaves a
remainder - 2 when divided by (x - 1). Find the
values of a and b.
please ans this as fast as possible​

Answer 1

GIVEN:-

• f (x) = ax^3 + x^2 - bx + 4 leaves a remainder 10 when divided by (x-2).

• The polynomial g (x) = ax^3 - 3x^2 + bx - 8 leaves aremainder - 2 when divided by (x - 1).

TO FIND:-

• Find the values of a and b.

THEOREM USED:-

• ${\boxed{\rm{\blue{Factor\:theorem}}}}$

Now,

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

$\implies\rm{x = 2}$

Now Put the value of x in f(x).

$\implies\rm{ax^3 + x^2 - bx + 4 = 10}$

$\implies\rm{a(2)^3 + (2)^2 - b(2) +4 = 10}$

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

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

$\implies\rm{ 8a - 2b = 2}$

$\implies\rm{2(4a - b) = 1}$

$\implies\rm{ 4a - b = 1}$.........1

Now, again

$\implies\rm{x - 1 = 0}$

$\implies\rm{ x = 1}$

Now Put the value of x in f(x).

$\implies\rm{ax^3 - 3x^2 + bx - 8 = -2}$

$\implies\rm{a(1)^3 - 3(1)^2 + b(1) -8 = -2}$

$\implies\rm{ a - 3 + b -8 = -2}$

$\implies\rm{ a + b -11 = -2}$

$\implies\rm { a + b = -2 + 11}$

$\implies\rm{ a + b = 9}$......2

Adding the equation 1 and 2.

$\implies\rm{ 4a - b + a + b = 9+1}$

$\implies\rm{ 5a = 10}$

$\implies\rm{ a =\dfrac{10}{5}}$

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

Now, Putting the value of a in equation 2.

$\implies\rm{ a + b = 9}$

$\implies\rm{ 2 + b = 9}$

$\implies\rm{ b = 9-2}$

$\implies\rm{ b = 7}$

Hence, The Value of a and b is 2 and 7 Respectively.