Find k, so that x2 + 2x + k is a factor of 2x4 + x3 - 14x2 + 5x + 6. Also, find all the zeroes of the two polynomials.
let g(x) = x2 + 2x + k and p(x) = 2x4 + x3 - 14x2 + 5x + 6
Then if g(x) is a factor of p(x)
By division algorithm
Dividend = (divisor) (quotient) + remainder
⇒ p(x) = g(x).q(x) + r(x)
Where q(x) is the quotient and
r(x) is the remainder which will be equal to zero.
i.e. r(x) = 0
Clearly q(x) = 2x2 - 3x - 8 - 2k and r(x) = (21 + 7k)x + (2k2 + 8k + 6) = 0
By comparing the coefficients, 21 + 7k = 0 and 2k2 + 8k + 6 = 0
2k2 + 8k + 6 = 0
k2 + 4k + 3 = 0
k2 + (k + 3k) + 3 = 0
k (k + 1) + 3(k + 1) = 0
(k + 1)(k + 3) = 0
⇒ k = - 1, - 3
Also, 21 + 7k = 0
Case1: k = - 1
21 + 7(- 1) = 0
= 21 - 7 = 14≠0
Hence k = - 1 is rejected.
Case2: k = - 3
21 + 7(- 3) = 0
= 21 - 21 = 0
∴ the value of k is - 3
g(x) = x2 + 2x - 3
x2 + 2x - 3 = 0
x2 + (- x + 3x) - 3 = 0
x(x - 1) + 3(x - 1) = 0
(x - 1)(x + 3) = 0
⇒ x = 1, - 3
q(x) = 2x2 - 3x - 8 - 2(- 3)
= 2x2 - 3x - 2
2x2 - 3x - 2 = 0
2x2 - (4x - x) - 2 = 0
2x(x - 2) + (x - 2) = 0
(x - 2)(2x + 1) = 0
⇒ x = 2, - 1/2
Now, we know g(x) and q(x) are factors of p(x)
∴ the zeroes of g(x) and q(x) will be the zeroes of p(x)
Hence, the zeroes of p(x) = - 3, - 1/2, 1, 2