Question

If f(x) = (x+2)(x+4)(x+6)(x+8) + k is a perfect square then k =

Answer 1

This type of question requires substituting.

(As we get 4th degree polynomial. We should use discriminant.)

We can expand the polynomial with having same quotient of x.

(x+2)(x+8) * (x+4)(x+6)

(x²+10x+16) * (x²+10x+24)

Now, as some part is repeated, we can let that 'T'.

(T+16) * (T+24) +k

T²+40T+(384+k)

This is a perfect square, then what is the discriminant value? 0.

D/4 = 400 -384 -k = -k +16 ∴ -k +16 =0

Therefore, k =16.