Find all roots and write the linear factorization

Question

f(x)=2x⁴-9x³+10x²+x-2

Answer 1

I started out graphing this equation. I could find 2 real roots that way, 2 and 1/2. To find the remaining, use synthetic division as the first step with the 2 known zeros.

2 | 2 -9 10 1 -2

4 -10 0 2

1/2| 2 -5 0 1 0

1 -2 -1

2 -4 -2

The remaining expression is 2x²-4x-2 These zeros you can find using the quadratic equation

[-(-4)±√(-4)2-4(2)(-2)]/[2(2)] Simplify

(4±√32)/4 Simplify

(4±4√2)/4 Simplify

x=1±√2

Now, to write these as factors

y=(x-2)(2x-1)(x-1+√2)(x-1-√2)