Polynomials of Higher Degree Revision Notes for VCE SSCE Mathematical Methods

Polynomials of Higher Degree

What is a quartic function?

A quartic function is a polynomial of degree 4. You can recognise these functions because the highest power of $x$ is 4.

The standard form of a quartic function is:

f(x) = ax^4 + bx^3 + cx^2 + dx + e

where $a \neq 0$

The coefficient $a$ must not equal zero, otherwise the $x^4$ term disappears and the function would no longer be quartic. The other coefficients ( $b$ , $c$ , $d$ , and $e$ ) can be any real numbers, including zero.

What does a quartic function look like?

Quartic functions can take many different shapes. Their graphs can have various numbers of turning points and can intersect the x-axis in different ways. Here are some common examples:

These graphs show the variety of shapes that quartic functions can produce:

$f(x) = x^4$ creates a smooth U-shaped curve (similar to a parabola but steeper at the edges)
$f(x) = x^4 - x^2$ creates a W-shaped curve with two valleys
$f(x) = (x-1)^2(x+2)^2$ touches the x-axis at two points but doesn't cross it
$f(x) = (x-1)^3(x+2)$ shows mixed behavior - crossing at one point and having a steeper approach at another

General properties of polynomial functions

When working with polynomials of any degree, there are important properties to understand about their graphs.

Number of x-intercepts

chatImportant

For a polynomial $P(x)$ of degree $n$ , there are at most n solutions to the equation $P(x) = 0$ . This means the graph of $y = P(x)$ can cross or touch the x-axis a maximum of $n$ times.

For example:

A quartic function (degree 4) can have 0, 1, 2, 3, or 4 x-intercepts
A cubic function (degree 3) can have 1, 2, or 3 x-intercepts
A quadratic function (degree 2) can have 0, 1, or 2 x-intercepts

Even vs odd degree polynomials

The degree of a polynomial determines whether its graph must cross the x-axis:

chatImportant

Even degree polynomials (like quartics, degree 4) may have no x-axis intercepts at all. For example, $P(x) = x^2 + 1$ never touches the x-axis because $x^2 + 1$ is always positive.

Odd degree polynomials (like cubics, degree 3) must have at least one x-axis intercept. The graph has to cross the x-axis somewhere, though it might cross multiple times.

This difference occurs because even degree polynomials can have the same end behavior on both sides (both ends going up or both going down), while odd degree polynomials must have opposite end behaviors (one end up, one end down).

Working with quartic expressions

The same techniques you learned for cubic functions work for quartic functions and other higher degree polynomials.

Creating sign diagrams

infoNote

A sign diagram helps you understand where a function is positive or negative. To create one for a quartic expression:

Find all the x-values where the expression equals zero (the zeros or roots)
Mark these values on a number line
Test the sign of the expression in each interval between these values
Record whether the expression is positive (+) or negative (-) in each region

Factorising quartic expressions

You can factorise quartic expressions using:

The factor theorem: If $P(a) = 0$ , then $(x-a)$ is a factor of $P(x)$
Polynomial division: Once you find one factor, divide to find the remaining factors
Pattern recognition: Sometimes you can spot factorisable patterns

Worked example: Sign diagrams for quartic expressions

lightbulbExample

Worked Example: Creating Sign Diagrams

Let's create sign diagrams for two different quartic expressions.

Part a: $(2-x)(x+2)(x-3)(x-5)$

This expression is already factorised, so we can identify the zeros directly.

The zeros occur when each factor equals zero:

$2-x = 0$ gives $x = 2$
$x+2 = 0$ gives $x = -2$
$x-3 = 0$ gives $x = 3$
$x-5 = 0$ gives $x = 5$

Now test the sign in each interval:

For $x < -2$ : All factors are negative except $2-x$ , giving positive overall

For $-2 < x < 2$ : The factor $(x+2)$ becomes positive, changing the sign to negative

For $2 < x < 3$ : The factor $(2-x)$ becomes negative, changing the sign to positive

For $3 < x < 5$ : The factor $(x-3)$ becomes positive, changing the sign to negative

For $x > 5$ : The factor $(x-5)$ becomes positive, making all factors contribute to positive

The sign diagram shows positive, negative, positive, negative, positive as we move from left to right.

Part b: $x^4 + x^2 - 2$

This expression needs factorising first. Let's use the factor theorem.

Test $P(1)$ :

P(1) = 1^4 + 1^2 - 2 = 1 + 1 - 2 = 0

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

Using polynomial division to find the other factor:

x^4 + x^2 - 2 = (x-1)(x^3 + x^2 + 2x + 2)

Now factorise $x^3 + x^2 + 2x + 2$ by grouping:

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

Therefore:

P(x) = (x-1)(x+1)(x^2+2)

The factor $(x^2+2)$ is always positive (since $x^2 \geq 0$ , so $x^2 + 2 \geq 2$ ).

The zeros are at $x = -1$ and $x = 1$ .

For $x < -1$ : Both $(x-1)$ and $(x+1)$ are negative, giving positive overall

For $-1 < x < 1$ : $(x+1)$ becomes positive while $(x-1)$ stays negative, giving negative overall

For $x > 1$ : Both factors are positive, giving positive overall

The sign diagram shows positive, negative, positive.

Worked example: Sketching a quartic function

lightbulbExample

Worked Example: Sketching a Quartic Graph

Let's sketch the graph of $p(x) = x^4 - 2x^2 + 1$ .

Step 1: Factorise the expression

Notice that this looks like a quadratic in $x^2$ :

p(x) = (x^2)^2 - 2(x^2) + 1

This is a perfect square:

p(x) = (x^2 - 1)^2

We can factorise further:

p(x) = [(x-1)(x+1)]^2

p(x) = (x-1)^2(x+1)^2

Step 2: Find the intercepts

x-intercepts: Set $p(x) = 0$

(x-1)^2(x+1)^2 = 0

This gives $x = 1$ and $x = -1$

Note that both factors are squared, so the graph touches the x-axis at these points but doesn't cross it.

y-intercept: Set $x = 0$

p(0) = 0^4 - 2(0)^2 + 1 = 1

The y-intercept is at $(0, 1)$

Step 3: Consider the shape

Since both $(x-1)^2$ and $(x+1)^2$ are squared factors, the function doesn't change sign at $x = -1$ or $x = 1$ . The graph touches the x-axis at these points and bounces back.

For large positive or negative values of $x$ , the $x^4$ term dominates, so the graph rises steeply on both sides.

Step 4: Sketch the graph

The graph shows:

x-intercepts at $(-1, 0)$ and $(1, 0)$ where the curve touches but doesn't cross
y-intercept at $(0, 1)$
The characteristic W-shape of a quartic with two repeated roots
Both ends of the graph rising towards positive infinity

Key features to remember when sketching quartics

When you're asked to sketch a quartic function in factorised form $y = a(x-\alpha)(x-\beta)(x-\gamma)(x-\delta)$ , follow these steps:

infoNote

1. Find the y-axis intercept

Set $x = 0$ and calculate the y-value.

2. Find the x-axis intercepts

Set each factor equal to zero and solve. These are the points where the graph crosses or touches the x-axis.

3. Create a sign diagram

This helps you understand where the function is positive or negative.

4. Consider end behavior

For a quartic with positive leading coefficient ( $a > 0$ ):

As $x$ increases to the right of all intercepts, $y$ becomes large and positive
As $x$ decreases to the left of all intercepts, $y$ becomes large and positive

For a quartic with negative leading coefficient ( $a < 0$ ):

Both ends go towards negative infinity

5. Check for repeated factors

If there's a repeated factor raised to an even power, the graph touches the x-axis at that point but doesn't cross it. The y-values have the same sign immediately to the left and right of that intercept.

If a factor is raised to an odd power, the graph crosses the x-axis at that point.

bookmarkSummary

Key Points to Remember:

A quartic function has the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ where $a \neq 0$
A polynomial of degree $n$ has at most n x-axis intercepts
Even degree polynomials may have no x-axis intercepts, but odd degree polynomials must have at least one
Quartic functions can have zero, one, two, three, or four x-intercepts
When a factor is repeated to an even power, the graph touches the x-axis without crossing (the sign stays the same on both sides)
Use the factor theorem and polynomial division to factorise quartic expressions
Sign diagrams help you understand the behavior of the function in different intervals

Polynomials of Higher Degree (VCE SSCE Mathematical Methods): Revision Notes