Question

If f(x) = x³+ax+b is divisible by (x-21)² then
the remainder obtained when f(x) is divided
by x+42 is ______
​

Answer 1

Step-by-step explanation:

Given If f(x) = x³+ax+b is divisible by (x-21)² then the remainder obtained when f(x) is divided  by x+42

• Now we have f (x) = x^3 + ax + b = (x – 21)^2 (x + a) [ (x – 21)^2 is a quadratic function and f(x) is a cubic function)
•                     So we get  
•                                    So x^3 + ax + b = (x^2 – 42x + 441) (x + a)
•                                     So x^3 + ax + b = x^3 – 42 x^2 + 441 x + ax^2 – 42 x a + 441 a
•                                   So x^3 + ax + b = x^3 + (a – 42) x^2 + (441 – 42 a) x + 441 a
•                                  Now comparing we have a – 42 = 0 since coefficient of x^2 is 0
•                                     So a = 42
•                                 Now we have the factors as  
•                                    So x^3 + ax + b = (x – 21)^2 (x + 42)
•                                  Or f(- 42) = (- 42 – 21)^2 (-42 + 42)
•                                      Or f(- 42) = 0
• Therefore the remainder will be zero.

Reference link will be

https://brainly.in/question/1314994