Question

Which of the following quartic functions has x=-1 and x = -3 as its only two real zeros?

y=x^4 - 4x^3 - 4x^2 - 4x - 3

y = -x^4 + 4x^3 + 4x^2 + 4x + 3

y x^4 + 4x^3 + 3x^2 + 4x

x^4 + 4x^3 + 4x^2 + 4x + 3

Answer 1

`y=x^4 + 4x^3 + 4x^2 + 4x + 3`

Explanation:

Consider function
`y=x^4 + 4x^3 + 4x^2 + 4x + 3.`

Factor it

`y=x^4 + 4x^3 + 4x^2 + 4x + 3\\ \\=x^4+x^3+3x^3+3x^2+x^2+x+3x+3\\ \\=x^3(x+1)+3x^2(x+1)+x(x+1)+3(x+1)\\ \\=(x+1)(x^3+3x^2+x+3)\\ \\=(x+1)(x^2(x+3)+(x+3))\\ \\=(x+1)(x+3)(x^2+1)`

This function has two real zeros
`x=-1` and
`x=-3` and two comples zeros (because
`x^2+1` cannot be factored futher).

Hence, the function
`y=x^4 + 4x^3 + 4x^2 + 4x + 3` has
`x=-1` and
`x=-3` as its only two real zeros.