a x^{3}+b x^{2}+c x+d is a polynomial on real x over real coefficients a, b, c, d wherein a \neq 0. Which of the following statements is true?

Correct Answer is : d can be chosen to ensure that x=0 is a root for any given set a,b,c.

d can be chosen to ensure that x=0 is a root for any given set a, b, c.

c alone cannot ensure that all roots are real.

a, b, c, d can be chosen to ensure that all roots are complex.

No choice of coefficients can make all roots identical.

GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 11 of 65

Marks: +1, -0

a x^{3}+b x^{2}+c x+d

is a polynomial on real

x

over real coefficients

a, b, c, d

wherein

a ≠

0. Which of the following statements is true?

Solution:

a x^{3}+b x^{2}+c x+d

Let

a=1, b=2, c=2, d=1

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

x=0

is a solution

f(0)=0

implies that

d=0

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