Question

Find the value of k that x² + 2x + k is a factor of
${2x}^{4} + {x}^{3} - 14 {x}^{2} + 5x \: + 6$
and also the two zeroes of the polynomial.​

Answer 1

Answer:

Step-by-step explanation:

$2x^4+x^3-14x^2+5x+6\\\\=2x^4+3x^3-11x^2-6x-2x^3-3x^2+11x+6\\\\=x(2x^3+3x^2-11x-6)-1(2x^3+3x^2-11x-6)\\\\=(x-1)(2x^3+3x^2-11x-6)\\\\=(x-1)(2x^3+7x^2+3x-4x^2-14x-6)\\\\=(x-1)(x(2x^2+7x+3)-2(2x^2+7x+3))\\\\=(x-1)(x-2)(2x^2+7x+3)\\\\=(x-1)(x-2)(2x^2+6x+x+3)\\\\=(x-1)(x-2)(2x(x+3)+1(x+3)))\\\\=(x-1)(x-2)(x+3)(2x+1)\\\\=(x-1)(x+3)(x-2)(2x+1)\\\\=(x(x+3)-(x+3))(x-2)(2x+1)\\\\=(x^2+2x-3)(x-2)(2x+1)$

Therefore k=-3

Hope it helps