Let real numbers `x_1, x_2, x_3, x_4, x_5` satisfy `x_(1+) x_2 + x_(3+)x_(4+) x_5=8` and `x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=16,` then

Consider the system of equations x_1+x_(2)^(2)+x_(3)^(3)+x_(4)^(4)+x_(5)^(5)=5 and x_1+2x_2+3x_3+4x_4+5x_5=15 where x_1,x_2,x_3,x_4,x_5 are positive real numbers. Then numbers of (x_1,x_2,x_3,x_4,x_5) is ___________.

Consider the system of equations x_1+x_(2)^(2)+x_(3)^(3)+x_(4)^(4)+x_(5)^(5)=5 and x_1+2x_2+3x_3+4x_4+5x_5=15 where x_1,x_2,x_3,x_4,x_5 are positive real numbers. Then numbers of (x_1,x_2,x_3,x_4,x_5) is ___________.

Consider the system of equations x_1+x_(2)^(2)+x_(3)^(3)+x_(4)^(4)+x_(5)^(5)=5 and x_1+2x_2+3x_3+4x_4+5x_5=15 where x_1,x_2,x_3,x_4,x_5 are positive real numbers. Then numbers of (x_1,x_2,x_3,x_4,x_5) is ___________.

Let the equation x^(5) + x^(3) + x^(2) + 2 = 0 has roots x_(1), x_(2), x_(3), x_(4) and x_(5), then find the value of (x_(2)^(2) - 1)(x_(3)^(2) - 1)(x_(4)^(2) - 1)(x_(5)^(2) - 1).

Divide x^4-x^3-3x^2+4x+5 by x^2-x+1

If x_1 x_2 x_3 = 4(4 + x_1 + x_2 + x_3 ) , then what is the value of [1//(2 + x_1 )] + [1//(2 + x_2 )] + [1//(2 + x_3 )] ?

The HCF of x^4 -11x^2 +10, x^2 – 5x+4 and x^3 – 3x^2 + 3x-1 is