Question

Find all the zeros of 2x4+x3-14x2-19x-6 two of its zeros are -2 and -1

Answer 1

`-2,\ -1,\ -(1)/(2),\ 3.`

Explanation:

Consider polynomial
`2x^4+x^3-14x^2-19x-6.`

If x=-2 is its zero, then you can divide the polynomial
`2x^4+x^3-14x^2-19x-6` by
`x+2` and get

`2x^4+x^3-14x^2-19x-6=(x+2)(2x^3-3x^2-8x-3).`

If x=-1, then the polynomial
`2x^4+x^3-14x^2-19x-6` can be rewritten as

`2x^4+x^3-14x^2-19x-6=(x+2)(x+1)(2x^2-5x-3).`

The quadratic polynomial has roots

`x_(1,2)=(-(-5)\pm√((-5)^2-4\cdot2\cdot(-3)))/(2\cdot 2)=(5\pm √(25+24))/(4)=(5\pm √(49))/(4)=3,-(1)/(2).`

Then the polynomial
`2x^4+x^3-14x^2-19x-6` has zeros
`-2,\ -1,\ -(1)/(2),\ 3.`