Question

If n is the degree of the polynomial, ${\left[\frac{2}{\sqrt{5x^2+1}-\sqrt{5x^3-1}}\right]}^8+{\left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]}^8$ and m is the coefficient of $x^n$ in it, then the ordered pair (n, m) is equal to:

• A. $(24,(10{)}^8)$
• B. $(8,5(10{)}^4)$
• C. $(12,(20{)}^4)$
• D. $(12,8(10{)}^4)$

Answer 1

Answer: (C)

${\left[\frac{2\left(\sqrt{5x^{3}1}\sqrt{5x^{3}1}\right)}{2}\right]}^8+{\left[\frac{2\left(\sqrt{5x^3+1}-\sqrt{5x^3-1}\right)}{2}\right]}^8$
$=2\left[{}^8{C}_0{\left(\sqrt{5x^3+1}\right)}^8{+}^8{C}_2{\left(\sqrt{5x^3+1}\right)}^8\left(5x^3-1\right){+}^8{C}_4{\left(\sqrt{5x^3+1}\right)}^4{\left(5x^3-1\right)}^2{+}^8{C}_8{\left(\sqrt{5x^3+1}\right)}^2{\left(5x^3-1\right)}^3{+}^8{C}_8{\left(5x^3-1\right)}^4\right]$
$=2\left[{\left(5x^3-1\right)}^4+28{\left(5x^3-1\right)}^3\left(5x^3-1\right)+70{\left(5x^3-1\right)}^2+28\left(5x^3-1\right){\left(5x^3-1\right)}^3+{\left(5x^3-1\right)}^4\right]$
$h=12\&m=2\left(5^4+140.5^3+70.5^4+140.5^3+5^4\right)$
$=160000={\left(20\right)}^4$