Question

If one of the roots of equation \(ax^4+bx^3+cx^2+dx+e=0\) is \(\sqrt{2}+\sqrt{-3}\), then arrange the following in non-decreasing order \((a\neq 0)\). A. \(a\),
B. \(b\),
C. \(c\),
D. \(d\),
E. \(e\).

• A. B, D, A, E, C
• B. D, B, A, E, C
• C. B, A, D, C, E
• D. B, D, A, C, E

Hint:

When a polynomial has real rational coefficients, imaginary and irrational conjugate roots must also be considered.

Answer 1

The Correct Option is D

Concept:
If a polynomial has real rational coefficients and one root is \(\sqrt{2}+i\sqrt{3}\), then its conjugate and related irrational conjugates also occur as roots.

Step 1: Write the given root. \[ \sqrt{2}+\sqrt{-3}=\sqrt{2}+i\sqrt{3} \] The related roots are: \[ \sqrt{2}+i\sqrt{3},\quad \sqrt{2}-i\sqrt{3},\quad -\sqrt{2}+i\sqrt{3},\quad -\sqrt{2}-i\sqrt{3} \]

Step 2: Form the polynomial. \[ \left[(x-\sqrt{2})^2+3\right]\left[(x+\sqrt{2})^2+3\right] \] \[ =\left[x^2-2\sqrt{2}x+5\right]\left[x^2+2\sqrt{2}x+5\right] \] Using \((A-B)(A+B)=A^2-B^2\), \[ =(x^2+5)^2-(2\sqrt{2}x)^2 \] \[ =x^4+10x^2+25-8x^2 \] \[ =x^4+2x^2+25 \]

Step 3: Compare with \(ax^4+bx^3+cx^2+dx+e=0\). \[ a=1,\quad b=0,\quad c=2,\quad d=0,\quad e=25 \]

Step 4: Arrange in non-decreasing order. \[ b=0,\quad d=0,\quad a=1,\quad c=2,\quad e=25 \] So, \[ B,D,A,C,E \] \[ \therefore \text{Correct Answer is (D)} \]