Methods2025 Exam 1 · Q7Tech-free6 marks

Factorising a cubic and its product with a linear function

Let $f: \R \to \R, f(x) = x^3 - x^2 - 16x - 20$ .

Verify that $x = 5$ is a solution of $f(x) = 0$ .

1 mark

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Express $f(x)$ in the form $(x+d)^2(x-5)$ , where $d \in \R$ .

2 marks

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Consider the graph of $y = f(x)$ , as shown below.

Complete the coordinate pairs of all axial intercepts of $y = f(x)$ .

Graph of the cubic y = f(x). It touches the negative x-axis at a turning point and has a local minimum to the right of the y-axis. It cuts the positive x-axis further right. The intercepts are labelled with blank coordinate pairs ( , 0), (0, ) and ( , 0) for students to complete.

1 mark

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d.i.

Let $g: \R \to \R, g(x) = x + 2$ .

State the coordinates of the stationary point of inflection for the graph of $y = f(x)g(x)$ .

1 mark

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d.ii.

Write down the values of $x$ for which $f(x)g(x) \geq 0$ .

1 mark

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