5 Find the integer values of k for which the given equation

5.)

Find the integer value(s) of k for which the given equation has rational roots, and find the roots.

x³ + kx² + kx+ 2 = 0

Solution

As per the Rational Roots theorem, if P(x) is a polynomial with integer coefficients and if p/q is a zero of P(x), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x). Here, the only factors of 2 are ± 1 and ± 2. Therefore, if P(x) = x³+kx²+kx+2 has rational roots, these are either ± 1 or ± 2.

In view of the above, x³ + kx² + kx + 2 has rational roots if k = 3.In this case, -2 is a root i.e. x+2 is a factor of x³ + kx² + kx + 2   if k = 3. On substituting k = 3 in the given expression, we get P(x) = x³+3x² +3x +2 = x³+2x² + x² +2x+x +2 = x²(x+2) +x(x+2)+1(x+2) = (x+2)(x² +x +1). On using the quadratic formula, the roots of x²+x+1 are [-1±{(1)²-4*1*1}]/2 or, [-1±-3]/2. These are complex roots.

Thus, P(x) = x³+kx²+kx+2 has one rational root i.e. -2 if k = 3.