Question

Find the value of ' a ' for which the polynomial

${x}^{5} - a {x}^{2} - ax + 1$

has −1 as a root with

multiplicity at least 2.​

Answer 1

The value of 'a' for which the polynomial x^{5} -ax^{2} -ax+1 has −1 as a root with  multiplicity at least 2 is (-5).

• For having -1 as root with multiplicity at least 2, the -1 must be the root of the polynomial as well as root of the derivative of the polynomial.
• f(x) = $x^{5} -ax^{2} -ax+1$ . Now f(-1) = 0. This is the trivial case by putting this value we can see. That is -1 - a + a +1 = 0
• Now $f^{'}(x) = 5x^{4} -2ax-a$ .
• Now $f^{'}(-1) = 0$ . Putting this we can see 5 +2a - a = 0
• This implies a = -5
• So for a = -5, the polynomial will have -1 as a root with multiplicity at least 2.