Question

Find k so that x2 + 2x + k is a factor of 2x4 + x3 –14 x2 + 5x + 6. Also find all the zeroes of the two polynomials.​

Answer 1

Answer:

If g(x)=x2 + 2x + k is a factor of f(x) = 2x4 + x3 - 14x2 + 5x + 6

Then remainder is zero when f(x) is divided by g(x)

Let quotient =Q and remainder =R

Let us now divide f(x) by gx)

R = x(7k + 21) + (2k2 + 8k + 6) -------(1)

and Q = 2x2 - 3x - 2(k + 4).------------(2)

⇒x (7k + 21) + 2 (k2 + 4k + 3) = 0

⇒7k + 21 = 0 and k2 + 4k + 3 = 0

⇒ 7(k + 3) = 0 and (k + 1) (k + 3) = 0

⇒ k + 3 = 0

⇒k = -3

Substituting the value of k in the divider x2 + 2x + k, we obtain: x2 + 2x - 3 = (x + 3) (x - 1) as the divisor.

Hence two zeros of g(x) are -3 and 1.------(3)

Putting k=-3 in (2) we get

Q = 2x2 - 3x - 2

= 2x2 - 4x + x - 2

= 2x(x - 2) + 1 (x - 2)

= (x - 2)(2x + 1)

Q=0 if x-2=0 or 2x+1=0

So other two zeros of f(x) are 2 and -12-------(4)

As g(x) is a factor of f(x) so zeros of G(x) are zeros of f(x) also

Hence from (3) and (4) we get

The zeros of f(x) are: -3 ,1,, 2 and 12