Let R1 and R2 are the remainders when the polynomials x^3 + 2x^2 - 5ax - 7 and x^3 + ax^2 - 12x + 6 are divided by x + 1 and x - 2 respectively. If R1+R2 = 6, find the value of a.

2.44

Step-by-step explanation:

Given: x³ + 2x² - 5ax - 7 and x³ + ax² - 12x + 6

Also, R1 + R2 = 6

in order to find the value of a:

Let p (x) = x³ + 2x² - 5ax - 7 and q (x) = x³ + ax² - 12x + 6

Using remainder theorem i. e if a polynomial p (x) is divisible by polynomial of form x - a then remainder is given by p (a).

Then,

R1 = p (-1) = (-1) ³ + 2 (-1) ² - 5a (-1) - 7 = - 1 + 2 + 5a - 7 = 5a - 6

R2 = q (2) = 2³ + a (2) ² - 12 (2) + 6 = 8 + 4a - 24 + 6 = 4a - 10

Now,

R1 + R2 = 6

5a - 6 + 4a - 10 = 6

9a = 22

a=2.44

Therefore, Value of a is 2.44