Question

Solve the equation 2x^3 – 5x² + x + 2 = 0 given that 2 is a zero of f (x) = 2x^3 – 5x^2 + x +2.

Answer 1

`f(x)=2x^3-5x^2+x+2`
`f(2)=2(2)^3-5(2)^2+(2)+2=16-20+2+2=0`

Hence, 2 is a zero of f(x). That is x - 2 is a factor of f(x).

So we can find

`(2x^3-5x^2+x+2)/(x-2)`
`\Rightarrow(2x^3-5x^2+x+2)/(x-2)=2x^2-x-1`
`\text{Next we solve }2x^2-x-1=0`
`\begin{gathered} \Rightarrow2x^2-x-1=0 \\ 2x^2+x-2x-1=0 \\ x(2x+1)-1(2x+1)=0 \\ \Rightarrow(x-1)(2x+1)=0 \\ \Rightarrow x-1=0\text{ or 2x+1=0} \\ \Rightarrow x=1\text{ or 2x=-1} \\ x=1\text{ or x =-}(1)/(2) \end{gathered}`

Hence,

`x=2,1,\text{ or -}(1)/(2)`