THEORY OF EQUATIONS PROBLEMS WITH SOLUTIONS

Problem 1 :

A zero of x³ + 64 is

(1) 0     (2)  4    (3)  4i     (4)  -4

Solution

Problem 2 :

If f and g are polynomials of degree m and n respectively and if h(x) = (f o g)(x), then the degree of h is

(1) mn     (2)  m+n   (3)  mⁿ     (4) n^m

Solution

Problem 3 :

A polynomial equation in x of degree n always has 

(1)  n distinct roots      (2) n real roots    (3)  n complex roots

(4) at most one root

Solution

Problem 4 :

If α, β and γ are the zeros of x³ + px² + qx + r, then Σ 1/α is

(1)  -q/r      (2) -p/r    (3)  q/r     (4) -q/r

Solution

Problem 5 :

According to the rational root theorem, which number is not possible rational zero of

4x⁷ + 2x⁴ - 10x³ - 5 ?

(1)  -1      (2) 5/4    (3)  4/5     (4) 5

Solution

Problem 6 :

The polynomial x³ - kx² + 9x has three real zeros if and only if k satisfies

(1)  |k| ≤ 6        (2) |k| = 0    (3)  |k| > 6     (4) |k|  ≥ 6

Solution

Problem 7 :

The number of real numbers in [0, 2π] satisfying

sin⁴x - 2sin²x + 1 is

(1)  2       (2) 4    (3)  1     (4) ∞

Solution

Problem 8 :

If x³ + 12x² + 10ax + 1999 definitely has a positive zero, if and only if

(1)  a ≥ 0        (2) a > 0    (3)  a < 0     (4) a ≤ 0

Solution

Problem 9 :

The polynomial x³ + 2x + 3 has

(1) one negative and two imaginary zeros

(2)  one positive and two imaginary zeros

(3) three real zeros     (4)  no zeros.

Solution

Problem 10 :

The number of positive zeros of the polynomial

is

(1)  0        (2) n    (3)  < n     (4) r

Solution