Review question

# Can we sketch the graph $y=x^3-x^2-x+1$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6612

## Solution

A sketch of the graph $y=x^3-x^2-x+1$ appears on which of the following axes?

The coefficient of $x^3$ is positive, so for large $x$ we get a positive $y$, in which case (a) cannot be the graph.

When $x=0$, $y=1$, so the $y$-intercept is $+1$ and therefore (b) cannot be the graph.

We note that if $x=1$ then $y=0$, so the point $(1,0)$ is on the graph, which is not the case for (d).

Hence, the correct answer is (c).

Alternatively, we can factorise the graph into $y=(x+1)(x-1)^2$, using the Factor Theorem.

This curve cuts the $x$-axis at $-1$, and touches it at $1$. Hence, the correct answer is (c).