Question

The polynomials 2x³ – 7x² + ax – 6 and x³ – 8x² + (2a + 1)x – 16 leaves the same remainder when divided by x – 2. Find the value of ‘a’. 

Answer 1

Let f(x) = 2x³ – 7x² + ax – 6

x – 2 = 0 ${\displaystyle \Rightarrow }$ x = 2 

When f(x) is divided by (x – 2), remainder = f(2) 

∴ f(2) = 2(2)³ – 7(2)² + a(2) – 6 

= 16 – 28 + 2a – 6

= 2a – 18 

Let g(x) = x³ – 8x² + (2a + 1)x – 16 

When g(x) is divided by (x – 2), remainder = g(2) 

∴ g(2) = (2)³ – 8(2)² + (2a + 1)(2) – 16 

= 8 – 32 + 4a + 2 – 16 

= 4a – 38  

By the given condition, we have: 

f(2) = g(2) 

2a – 18 = 4a – 38 

4a – 2a = 38 – 18  

2a = 20 

a = 10 

Thus, the value of a is 10.