Solving Cubic Inequalities Revision Notes for VCE SSCE Mathematical Methods

Solving Cubic Inequalities

What are cubic inequalities?

A cubic inequality involves a polynomial of degree three and an inequality sign (such as $\leq$ , $<$ , $\geq$ , or $>$ ). Just like with quadratic inequalities, we can solve cubic inequalities by examining the graph of the corresponding polynomial function.

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The Key Approach:

To solve a cubic inequality, we follow three essential steps:

Sketch the graph of the cubic function
Identify where the graph satisfies the inequality condition
Express the solution using set notation

This is the same graphical approach used for quadratic inequalities, adapted for cubic functions.

Method for solving cubic inequalities

To solve a cubic inequality, follow these steps:

Step 1: Use the factor theorem

Find at least one linear factor by testing simple values (like $x = 1$ , $x = -1$ , $x = 2$ , etc.) in the polynomial. If P(a) = 0, then (x - a) is a factor.

Step 2: Complete the factorisation

Use polynomial division to find the remaining factors. Express the cubic in fully factored form.

Step 3: Find the intercepts

x-axis intercepts: Set each factor equal to zero and solve for the zeros or roots
y-axis intercept: Substitute $x = 0$ into the function

Step 4: Sketch the graph

Draw a smooth curve through the intercepts, remembering that:

Cubics with positive leading coefficient start low (on the left) and end high (on the right)
Repeated factors create turning points where the graph touches but doesn't cross the x-axis

Step 5: Solve the inequality from the graph

For $P(x) \leq 0$ or $P(x) < 0$ : find where the graph is on or below the x-axis
For $P(x) \geq 0$ or $P(x) > 0$ : find where the graph is on or above the x-axis

Step 6: Express the solution

Write your answer using set notation or interval notation.

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Reading Solutions from Graphs:

Remember the simple rule: "Less than means below, greater than means above"

$\leq$ or $<$ : Look for where the curve is below (or touching) the x-axis
$\geq$ or $>$ : Look for where the curve is above (or touching) the x-axis

Worked example

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Worked Example: Solving a Cubic Inequality with a Repeated Factor

Find $\{x : x^3 + x^2 - 5x + 3 \leq 0\}$

Solution:

Let $P(x) = x^3 + x^2 - 5x + 3$

First, use the factor theorem to find an initial factor by testing simple values:

P(1) = 1 + 1 - 5 + 3 = 0

Since P(1) = 0, we know that (x - 1) is a factor.

Now perform polynomial division to find the complete factorisation:

y = (x - 1)^2(x + 3)

Notice this cubic has a repeated factor (x - 1)². This is important for sketching the graph.

Next, find the axis intercepts:

x-axis intercepts: Set $y = 0$

$(x - 1)^2 = 0$ gives $x = 1$
$(x + 3) = 0$ gives $x = -3$

There are only two x-intercepts: $x = 1$ and $x = -3$

y-axis intercept: Set $x = 0$

P(0) = 0^3 + 0^2 - 5(0) + 3 = 3

The y-intercept is at $(0, 3)$

Now sketch the graph. The graph:

Passes through $(-3, 0)$ , $(0, 3)$ , and $(1, 0)$
Has a turning point at (1, 0) because of the repeated factor $(x - 1)^2$ — the graph touches the x-axis at $x = 1$ but doesn't cross it
Starts from the bottom left (as $x \to -\infty$ , $y \to -\infty$ )
Ends at the top right (as $x \to \infty$ , $y \to \infty$ )

From the graph, we can see that $y \leq 0$ in two regions:

When $x \leq -3$ (the graph is below the x-axis to the left of $x = -3$ )
When $x = 1$ (the graph touches the x-axis at exactly this point)

Therefore, the solution is:

\{x : x^3 + x^2 - 5x + 3 \leq 0\} = (-\infty, -3] \cup \{1\}

Note: The square bracket at $-3$ indicates that $-3$ is included in the solution (since we have $\leq$ not $<$ ). The single-element set $\{1\}$ shows that only the point $x = 1$ satisfies the inequality in that region.

Understanding repeated factors

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Critical Concept: How Repeated Factors Affect the Graph

When a cubic has a repeated factor, such as $(x - a)^2$ , the graph behaves differently at that x-intercept:

Single factor $(x - a)$ : the graph crosses the x-axis at $x = a$
Repeated factor $(x - a)^2$ : the graph touches the x-axis at $x = a$ but doesn't cross it — this creates a turning point

This is crucial when solving inequalities because a point where the graph just touches the x-axis may be an isolated solution point, as we saw in the example above where $\{1\}$ appeared as a separate part of the solution.

Key steps summary

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Summary: The Factor-Sketch-Solve Method

Factor the polynomial using the factor theorem and division
Find all intercepts to help sketch the graph accurately
Sketch the graph carefully, noting any repeated factors
Read the solution from the graph based on the inequality sign
Write the answer in proper set notation or interval notation

Remember!

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Key Points to Remember:

Cubic inequalities are solved by sketching the graph of the cubic function and identifying where it satisfies the inequality
Use the factor theorem to find factors: if $P(a) = 0$ , then $(x - a)$ is a factor
Repeated factors create turning points where the graph touches but doesn't cross the x-axis
For $\leq$ or $<$ : find where the graph is below (or on) the x-axis
For $\geq$ or $>$ : find where the graph is above (or on) the x-axis
Always express your final answer using correct set notation, such as $(-\infty, a] \cup \{b\}$ or $(c, \infty)$

Solving Cubic Inequalities (VCE SSCE Mathematical Methods): Revision Notes