# List all possible rational zeros. Find all real zeros of the polynomial and factor completely. Please show all of your work. f(x)=2x^4-13x^3-17x^2+145x+75

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Given `f(x)=2x^4-13x^3-17x^2+145x+75` :

(1) The possible rational roots are of the form `p/q` where `p` is a factor of 75 and `q` is a factor of 2. **Thus the list of possible rational roots is:**

`+-1,+-3,+-5,+-15,+-25,+-75,+-1/2,+-3/2,+-5/2,+-15/2,+-25/2,+-75/2`

(2) Using synthetic division (or polynomial long division) we find that -3 is a root (or (x+3) is a factor) so

`2x^4-13x^3-17x^2+145x+75=(x+3)(2x^3-19x^2+40x+25)`

We turn our attention to `2x^3-19x^2+40x+25`

(3) Using synthetic division or polynomial long division we find that 5 is a root (or (x-5) is a factor) so

`2x^4-13x^3-17x^2+145x+75=(x+3)(x-5)(2x^2-9x-5)`

The trinomial on the end factors: `2x^2-9x-5=(2x+1)(x-5)`

(4) **Thus the complete factorization is:**

`2x^4-13x^3-17x^2+145x+75=(x-5)^2(x+3)(2x+1)`