cubic intercepts

Practise cubic intercepts with an original VectorPaths question, worked explanation and related VCE Methods practice.

3 practice marks

Original practice question

Let $f(x)=\left(x - 4\right) \left(x - 1\right) \left(x + 2\right)$. Find all axis intercepts and describe the end behaviour needed for a sketch.

Worked explanation

• Set each factor to zero

  The distinct roots are $x=-2$, $x=1$, $x=4$. Each simple root crosses the x-axis.

• Substitute zero

  $f(0)=8$.

• Read the leading term

  The leading term is $x^3$: the left end tends to $-\infty$ and the right end to $\infty$.

x-intercepts: $(-2,0)$, $(1,0)$, $(4,0)$; y-intercept $(0,8)$.

Original VectorPaths practice; not an official VCAA question or marking rubric.