Question

If x + k is the GCD of x2 _ 2x _ 15 and x3

+ 27, find the value of k

Answer 1

I think the value of k is 3

Answer 2

Answer:

Values of k are -5 and 3

Step-by-step explanation:

Given that x + k is the GCD of $x^2 - 2x - 15\text { and }x^3+27$

we have to find the value of k.

As x + k is the GCD $x^2 - 2x - 15\text{and }x^3+27$

⇒ x=-k is the zero of given polynomials.

∴ By remainder theorem

When we substitute x=-k in both the polynomials the remainder we get is 0

$P(x)= x^2 - 2x - 15$

$P(-k)=(-k)^2-2(-k)-15=0$

$k^2+2k-15=0$

$k^2+5k-3k-15=0$

$k(k+5)-3(k+5)=0$

$(K+5)(k-3)=0$

⇒ k=-5 and k=3

When we substitute in other

$k^3+27=0$

we get complex values of k

Hence, values of k are -5 and 3