Question

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Answer 1

Answer:

a = 2

Step-by-step explanation:

(i) R₁ = Remainder when p(x) is divided by x + 1

Given p(x) = x² + 2x² - 5ax - 7.

∴ R₁ = f(-1)

      = (-1)³ + 2(-1)² - 5a(-1) - 7

      = -1 + 2 + 5a - 7

     = 5a - 6

(ii) R₂ = Remainder when q(x) is divided by x - 2:

Given q(x) is divided by x - 2.

∴ R₂ = q(2)

       = (2)³ + a(2)² - 12(2) + 6

      = 8 + 4a - 24 + 6

      = 4a - 10

(iii)

Given that 2R₁ + R₂ = 6.

⇒ 2(5a - 6) + (4a -10) = 6

⇒ 10a - 12 + 4a - 10 = 6

⇒ 14a = 28

⇒ a = 2.

Therefore, the value of a = 2.

Hope it helps!

Answer 2

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