Total-no-of-polynomials-of-the-form-x-3-ax-2-bx-c-that-are-divisible-by-x-2-1-where-a-b-c-1-2-3-10-is-1-10-2-15-3-5-4-8-

Question Number 32681 by rahul 19 last updated on 31/Mar/18

T o t a l n o . o f p o l y n o m i a l s o f t h e f o r m

x^{3} + {a x}^{2} + b x + c t h a t a r e d i v i s i b l e b y

x^{2} + 1, w h e r e a, b, c \in 1, 2, 3, \dots ., 10 i s

1) 10

2) 15

3) 5

4) 8

Commented by Rasheed.Sindhi last updated on 31/Mar/18

If x^{2} + 1 is one factor of x^{3} + {a x}^{2} + b x + c

then other factor will be linear and of

type x + p .

(x^{2} + 1) (x + p) = x^{3} + {p x}^{2} + x + p

Comparing with given polynomial

we have b = 1 & a = c = p

Hence the given polynomial will be

of the form :

x^{3} + {a x}^{2} + x + a

Since a \in 1, 2, 3, \dots, 10 (Ten values)

Hence total number of polynomials

of the given form with given conditions

is 10

Commented by rahul 19 last updated on 31/Mar/18

t h a n k u s i r!