Question

Let a, b, c, d be four integers (not necessarily distinct) in the set {1, 2, 3, 4, 5}. Then the
number of polynomials x4 + ax3 + bx2 + cx + d which is divisible by x + 1 is (A) between 55 and 65.
(B) between 66 and 85.
(C) between 86 and 105
(D) more than 105.

Answer 1

@ganeshie8 , check this one

Answer 2

\(\large 1-a+b-c+d = 0\)

Answer 3

stil have no clue lol

Answer 4

@ganeshie8 what does this prove by placing -1 in place of x in the quadratic.......

Answer 5

If \(x+1\) divides the polynomial evenly, then \(x=-1\) is a solution to the polynomial. So it has to satisfy the polynomial

Answer 6

\(\large a+c = b+d+1 \)
say \(a+c = y \\ b+d = x\)

Solve below in integers :
\(\large y = x+1 \\ 3 \le y \le 10; 2 \le x \le 9\)

Answer 7

@ganeshie8 okk.... thanxxx

Answer 8

np, it can be tricky... can u show me the solution once u work it ?

Answer 9

Number of ways for x :

x = 2 : (1, 1) = `1 way`
x = 3 : (1, 2), (2, 1) = `2 ways`
x = 4 : (1, 3), (2, 2), (3,1) = `3 ways`
x = 5 : (1, 4), (2, 3), (3, 2), (4, 1) = `4 ways`
x = 6 : (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) = `5 ways`
x = 7 : (2, 5), (3, 4), (4, 3), (5, 2) = `4 ways`
x = 8 : (3, 5), (4, 4), (5, 3) = `3 ways`
x = 9 : (4, 5), (5, 3) = `2 ways`

Answer 10

since \(y = x+1\),

total number of polynomials =

\(\large 1*2 + 2*3 + 3*4 + 4*5 + 5*4 + 4*3 + 3*2 + 2*1 = 80\)

Answer 11

let me knw if smthng doesnt make sense...

Answer 12

i think the last one 2*1 shud nt be der....

Answer 13

bdw answer is B

Answer 14

it has to be there, it represents the case `x = 9, y = 10`