Show that for any real numbers `lambda`, the polynomial `P(x)=x^7+x^3+lambda`, has exactly one real root.

Show that a polynomial of an odd degree has at least one real root.

Prove that the equation x^(3) + x – 1 = 0 has exactly one real root.

If a lt b lt c lt d , then for any real non-zero lambda , the quadratic equation (x-a)(x-c)+lambda(x-b)(x-d)=0 ,has real roots for

Prove that the equation 3x^(5)+15x-18=0 has exactly one real root.

If a lt b lt c lt d , then for any real non-zero lambda , the quadratic equation (x-a)(x-c)+lambda(x-b)(x-d)=0 ,has (a) no real roots. (b) one real root between a and c (c) one real root between b and d (d) Irrational roots.

Prove that the equation,3x^(3)-6x^(2)+6x+sin x=0 has exactly one real root.

find the values of lambda such that the equation tan^(2)x+lambda tan x+4=0 has atleast one real root

Consider the polynomial f(x)=1 + 2x + 3x^2 +4x^3 for all x in R So f(x) has exactly one real root in the interval

Find the set of values of lambda for which the equation |x^(2)-4|x|-12]=lambda has 6 distinct real roots.

If the cubic equation 3x^3 +px +5=0 has exactly one real root then show that p gt 0