Question

let p and q be the remainders , when the polynomials x3+2x2-5ax-7 and x3+ax2-12x+6 are divided by (x+1) and (x-2) respectively . if 2p+q=6 then find the value of a

Answer 1

REMAINDER THEOREM AND RESOLUTION

Answer 2

Answer:

Value of a is 2.

Step-by-step explanation:

Given polynomials,

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

R1 and R2 is remainder when polynomials divided by  x + 1  and x - 2

2p + q = 6

To find: Value of a.

Let, f(x) = x³ + 2x² - 5ax - 7

g(x) = x³ + ax² - 12x + 6

Using Remainder theorem which states that if a polynomial p(x) is divisible by polynomial of form x - a then remainder is given by p(a).

According tot remainder theorem,

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

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

Now,

2p + q = 6

2( 5a - 6 ) + 4a - 10 = 6

10a + 4a - 12 -10 = 6

14a = 28

a = 2

Therefore, Value of a is 2.