Given:
The equation of graph is given as,
[tex]y=x^3-2x^2+x[/tex]
The objective is to find x intercepts and y intercepts without using graph.
Explanation:
To find y intercept:
If the graph passes through the y axis, the value of x will be zero.
At x = 0,
[tex]\begin{gathered} y=0^3-2(0)^2+0 \\ y=0 \end{gathered}[/tex]
Thus, the y intercept is (0,0).
To find x intercept:
If the graph passes through the x axis, the value of y will be zero.
At y = 0,
[tex]0=x^3-2x^2+x[/tex]
By rearranging and grouping the above equation,
[tex]\begin{gathered} 0=x(x^2-2x+1) \\ 0=x(x^2-x-x+1) \\ 0=x\lbrack x(x-1)-1(x-1)\rbrack \\ 0=x(x-1)(x-1) \end{gathered}[/tex]
On solving the above equation by splitting into two terms,
[tex]\begin{gathered} x=0 \\ x-1=0\text{ or x=1} \end{gathered}[/tex]
Thus, the x intercepts are (0,0) and (1,0).
So, the obtained x intercept is (0,0), (1,0) and the y intercept is (0,0).
Hence, option (d) is the correct answer.
