Question

anyone please solve this question correctly​

Answer 1

Given:

☛ polynomial p(x) = x² + 2x + k is a factor of polynomial g(x) = 2x⁴ + x³ - 14x² + 5x + 6

To Find:

☛ value of k

☛ zeroes of the two polynomials

Solution:

☛ Divide g(x) by p(x), Since p(x) is a factor of g(x) so on dividing g(x) by p(x) , the remainder must be 0.

[ Refer to the attachment for division step ]

We get remainder as,

☛ r = (21 + 7k)x + 2k² + 8k + 6

Since, p(x) is a factor of g(x) so remainder must be zero.

Therefore,

☛ (21 + 7k)x + 2k² + 8k + 6 = 0

On comparing,

(21 + 7k)x = 0 ; 2k² + 8k + 6 = 0

☛ 2k² + 8k + 6 = 0

➜ 2k² + 6k + 2k + 6 = 0

➜ 2k ( k + 3) + 2(k + 3) = 0

➜ (k + 3)(2k + 2) = 0

k = -3 ; k = -1

Put -1 in (21 + 7k) = 0

We get to know that it doesn't satisfies this equation.

Now,

k = -3 satisfies (2k² + 8x + 6=0 ) and (21 + 7k)x = 0.

So, k = -3

Now,

Put value of k in p(x)

☛ p(x) = x² + 2x + k

➜ p(x) = x² + 2x - 3

➜ p(x) = x² + 3x - x - 3

➜ p(x) = x(x + 3) -1 (x + 3)

➜ p(x) = (x + 3)(x - 1)

Zeroes of polynomial p(x) are -3 and 1.

By Division algorithm,

g(x) = p(x) × q(x) + r

g(x) = p(x) × q(x) + 0

We observe that q(x) is a factor of g(x) so zeroes of q(x) will also be the factors of g(x),

Now,

☛ q(x) = 2x² - 3x -(8 + 2k)

➜ 2x² - 3x - 2 { since k = -3 }

➜ 2x² + x - 4x - 2

➜ x( 2x + 1 ) - 2( 2x + 1)

➜ (2x + 1)(x - 2)

Zeroes of polynomial q(x) is -1/2 and 2