Question

Let R1 and R2 be the remainder when the polynomials f(x) = x^3 +2ax^2 - 5x - 7 and g (x) = x^3 +x^2 - 12x + 6a are divided by (x + 1) and (x - 2) respectively. If 2R1 + 2R2 = 12. Find the value of a......
QUICK.!!!!!!!

Answer 1

Make x+1, zero of the polynomial, therefore x = -1
R1 =  -1+2a+5-7 
R1 = -3+2a

Now, make x-2 a zero of the polynomial, therefore x=2
R2 = -12+6a


therefore, R1+R2= 6 , (took 2 common)
substitte the values
-3+2a+6a-12=6
8a=21
a=21/8

Answer 2

Answer:

Value of a is 2.625.

Step-by-step explanation:

Given polynomials,

f(x) = x³ + 2ax² - 5x - 7       and  g(x) = x³ + x² - 12x + 6a

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

2 × R1 + 2 × R2 = 12

To find: Value of a.

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,

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

R2 = g( 2 ) = 2³ + 2² - 12(2) + 6a = 8 + 4 - 24 + 6a = 6a - 12

Now,

2 × R1 + 2 × R2 = 12

2( 2a - 3 ) + 2( 6a - 12 ) = 12

4a - 6 + 12a - 24 = 12

16a = 12 + 30

16a = 42

a = 2.625

Therefore, Value of a is 2.625.