Question

find the value of p for which the polynomial 2 x ki power 4 + px ki power 3 +4x ki power 2 + 2x + 1 is exactly divoded by x-1/2

Answer 1

i)  x2 – 2x – 8

          (ii)  4s2 – 4s + 1

          (iii)  6x2 – 3 – 7x

          (iv)  4u2 + 8u

          (v)  t2 – 15

          (vi)  3x2 – x – 4

Sol.  (i)  x2 – 2x – 8

                   We have p(x) = x2 – 2x – 8

                   = x2 + 2x – 4x – 8 = x (x + 2) – 4 (x + 2)

                   = (x – 4) (x + 2)

                   For p(x) = 0, we have

                   (x – 4) (x + 2) = 0

                   Either x – 4 = 0 ⇒ x = 4

                   or x + 2 = 0 ⇒ x = – 2

                   ∴ The zeroes of x2 2x – 8 are 4 and –2.

                   Now, sum of the zeroes 

                   

                   

                   Thus, relationship between zeroes and the coefficients in x2 – 2x – 8 is verified.

          (ii)  4s2 – 4s + 1

                   We have p(s) = 4s2 – 4s + 1

                   = 4s2 – 2s – 2s + 1 = 2S (2S – 1) –1 (2s – 1)

                   = (2s – 1) (2s – 1)

                   For p(s) = 0, we have,

                   

Now,