Question

If one of the zeroes of a cubic polyno-
mial of the form x3 + ax2 + bx + c is the
negative of the other, then
(A) a is of negative sign and band care of
positive sign
(B) bis of negative sign and a and care of
positive sign
(C) a and care of opposite signs and bis
of negative sign
(D) a and b are of opposite signs and c is
of positive sign
(E) None of these​

Answer 1

Step-by-step explanation:

(a) Let p(x) = x3 + ax2 + bx + c

Let a, p and y be the zeroes of the given cubic polynomial p(x).

∴ α = -1 [given]

and p(−1) = 0

⇒ (-1)3 + a(-1)2 + b(-1) + c = 0

⇒ -1 + a- b + c = 0

⇒ c = 1 -a + b …(i)

We know that,

αβγ = -c

⇒ (-1)βγ = −c [∴α = -1]

⇒ βγ = c

⇒ βγ = 1 -a + b [from Eq. (i)]

Hence, product of the other two roots is 1 -a + b.

Alternate Method

Since, -1 is one of the zeroes of the cubic polynomial f(x) = x2 + ax2 + bx + c i.e., (x + 1) is a factor of f{x).

Now, using division algorithm,

⇒x3 + ax2 + bx +c = (x + 1) x {x2 + (a – 1)x + (b – a + 1)> + (c – b + a -1)

⇒x3 + ax2 + bx + (b – a + 1) = (x + 1) {x2 + (a – 1)x + (b -a+ 1)}

Let a and p be the other two zeroes of the given polynomial, then