CMI-2021-DataScience-A: 2

Question

The roots of the polynomial $p(x)=x^{4}-2x^{3}-2x^{2}+8x-8$ are:

• A. $1, -1, 2, 2+3 i$
• B. $1+i, 1-i, 2, -2$
• C. $1, -1+i, 2, 2+3 i$
• D. $1+i, -1+i, 2, -2$

Comments

$x^4 – 2x^3 – 2x^2 + 8x – 8$

$x^3 (x – 2) -2 (x^2 -4x +4)$

$x^3 (x – 2) -2(x-2)^2$

$(x-2)(x^3 – 2(x-2))$

$(x-2)(x^3 – 2x+4 +4 -4)$

$(x-2)(x^3 +8 -2x-4)$

$(x-2)(x^3 +2^3 -2(x+2))$

$(x-2)((x+2)(x^2 +4-2x) -2(x+2))$

$(x-2)(x+2)(x^2 -2x +2)$

$(x-2)(x+2)(x- (1\pm i))$

Answer 1

Answer is option B

we will see why

Answer 2

the rule for imaginary root exists in conjugate pair means if (a+ib) then other is (a-ib)

only follow the (B) option.