Question

8. The polynomials ax + 3x2 – 3 and 2x3 - 5x
+ a, when divided by x - 4, leave the same
remainder in each case. Find the value of a.
​

Answer 1

Given :

• The polynomials ax³ + 3x² - 3 and 2x³ - 5x + a give the same remainder when divided with ( x - 4 )

To Find :

• The value of a.

Let us take f(x) = ax³ - 3x² - 3

The remainder is x - 4

⟹ x - 4 = 0

⟹ $\boxed{\sf{x=4}}$

______________

f(4) = a(4)³ + 3(4)² - 3

Let us take the second polynomial also.

f(x) = 2x³ - 5x + a

f(4) = 2(4)³ + 5(4) + a

________________

Let us make both the Polynomials equal now, as it is given that their remainder is the same.

⟹ a(4)³ + 3(4)² - 3 = 2(4)³ + 5(4) + a

⟹ 64a + 48 - 3 = 128 - 20 + a

⟹ 63a + 45 = 108

⟹ 63a = 108 - 45

⟹ 63a = 63

⟹ $\sf{a=\frac{63}{63}}$

⟹ $\boxed{\sf{\red{a=1}}}$