What partial fractions are for

A rational function can be written as a quotient of two polynomial functions, provided the denominator isn’t zero. A partial fraction decomposition is an algebraic identity that expresses one rational expression as a sum of simpler rational expressions.

In this part of the syllabus, the denominator has at most two distinct linear factors, while the numerator has a lower degree than the denominator. So, in practice, an expression such as

is split into the form

The symbol $\equiv$ is significant: it shows that both sides are equal for every value of $x$ for which they are defined, rather than for just one convenient value.

The key pattern to remember is how each denominator factor matches its partial fraction term. Partial fraction templates for one or two linear factors

Finding the constants

First, factorise the denominator fully into linear factors. Next, write the partial fraction form using unknown constants. For example,

Multiply both sides by the full denominator:

At this point, either compare the coefficients of $x$ and the constant terms or substitute values that make one bracket zero. Comparing coefficients is the safer method because it checks the entire identity. Substituting the roots of the denominator is faster and works perfectly well when the proposed form is definitely correct.

In this example, substituting $x=3$ gives $22=4B$, so $B=\frac{11}{2}$. Substituting $x=-1$ gives $2=-4A$, so $A=-\frac{1}{2}$. Hence

Telescoping sums and later use

Partial fractions often turn up because they simplify the next stage of a problem. In calculus, they rearrange integrands before integration. In sequences and series, they can produce a telescoping sum: a finite sum in which the middle terms cancel in pairs.

For instance,

Therefore

The method works because the algebraic form has changed, but the expression remains equivalent.

The reciprocal function

The reciprocal function maps every non-zero real number to its multiplicative inverse. We write it as

The domain is $\{x\in\mathbb{R}:x\neq 0\}$, while the range is $\{y\in\mathbb{R}:y\neq 0\}$.

An asymptote is a straight line that a graph gets arbitrarily close to, though the graph doesn’t have to meet it. For $f(x)=\frac{1}{x}$, $x=0$ is the vertical asymptote and $y=0$ is the horizontal asymptote. There are no intercepts with either axis. The value $x=0$ isn’t allowed, and $\frac{1}{x}$ can never be zero.

The reciprocal function is a self-inverse function: its inverse function is the function itself. If $y=\frac{1}{x}$, rearranging gives $x=\frac{1}{y}$, so applying the same rule takes you back to the original input. On the graph, this appears as symmetry in the line $y=x$.

Linear rational functions

A linear rational function is a rational function with linear polynomials in both the numerator and denominator. In this syllabus, it has the form

Unless the question says otherwise, take the largest possible real domain. The denominator gives the excluded value:

The vertical asymptote is the vertical line that the graph approaches near an excluded input value. For this function,

The horizontal asymptote is the horizontal line approached as $x$ becomes very large positive or very large negative. Because the numerator and denominator have the same degree, compare their leading coefficients:

Put these two equations on the sketch before working out the curve in detail. They provide the frame of reference. With a poor calculator window, one branch may even look as though it has disappeared.

Intercepts and sketches

An axis intercept is a point where the graph meets one of the coordinate axes. For

where the relevant values are defined, substitute $x=0$ to find the $y$-intercept:

so the point is $\left(0,\frac{b}{d}\right)$ when $d\neq 0$. To find the $x$-intercept, set the numerator equal to zero:

which gives the point $\left(-\frac{b}{a},0\right)$ when $a\neq 0$.

A clear sketch of a linear rational function needs to show both asymptotes and any intercepts with the axes. Its two branches lie in the regions shaped by the asymptotes. Even though the fraction contains linear expressions, the graph doesn’t behave like a straight line.

Transformations and mathematical representation

The graph of $f(x)=\frac{ax+b}{cx+d}$ can often be viewed as a transformation of $y=\frac{1}{x}$. Its asymptotes reveal the translation: the centre of the two-branch shape lies where the vertical and horizontal asymptotes intersect.

This also illustrates a wider idea about functions. The same relationship may be shown symbolically with a formula, visually with a graph, or numerically in a table. A formula gives exact asymptotes and intercepts, while a graph shows the shape and how the branches relate to each other. The notation and language of functions developed historically through several mathematical traditions. Mathematical knowledge is therefore stable in its logic, even though its forms of representation aren’t frozen.

The two forms in this part of the course

The rational functions in this section build on the linear-over-linear case from SL. You need to know two forms:

and

The reciprocal function is a special case in this family: $\frac{1}{x}$ comes from a constant numerator and a linear denominator.

The algebra determines the shape of the graph. Zeros of the denominator control vertical behaviour, while relative degrees determine end behaviour. Zeros of the numerator give the $x$-intercepts.

Functions of the form $\frac{ax+b}{cx^2+dx+e}$

Consider

where $c\neq 0$. Begin with the denominator. The real solutions of

are excluded $x$-values and usually give vertical asymptotes. Since the denominator is quadratic, the graph may have two, one, or no real vertical asymptotes.

The numerator has a smaller degree than the denominator, so the graph approaches the $x$-axis as $x$ becomes large in either the positive or negative direction. Its horizontal asymptote is therefore

To find the $x$-intercept, set the numerator equal to zero:

when $a\neq 0$ and the denominator is not zero there. Substitute $x=0$ for the $y$-intercept:

when $e\neq 0$.

Functions of the form $\frac{ax^2+bx+c}{dx+e}$

Now consider

where $d\neq 0$. Setting the denominator equal to zero gives the vertical asymptote

In this case, the numerator’s degree is one greater than the denominator’s. The graph therefore has an oblique asymptote: a non-horizontal straight line approached by the graph for large positive or negative $x$. Use polynomial division to find it.

For example, dividing $ax^2+bx+c$ by $dx+e$ gives

The oblique asymptote is

There’s no need to memorise a separate formula for $m$ and $n$. Division is safer and less prone to error. As the magnitude of $x$ becomes very large, the remainder term $\frac{k}{dx+e}$ becomes very small, so the graph approaches the line.

Find the $x$-intercepts by solving

provided the denominator is not zero at those roots. The $y$-intercept is

when $e\neq 0$.

It helps to compare the asymptote rules for the two AHL forms side by side. Comparison of asymptote and intercept rules for the two AHL rational-function forms.

Sketching accurately: algebra first, technology second

Use this order when working with these graphs:

1. State the domain restrictions from the denominator.
2. Find vertical asymptotes from real denominator zeros.
3. Use the degrees of the numerator and denominator to find the horizontal or oblique asymptote.
4. Find any intercepts with the axes.
5. Check the branch positions with a graphing package or GDC, then choose a sensible window.

Dynamic graphing software is useful for this work. Changing a parameter moves the asymptotes and may even alter how many vertical asymptotes are visible. Still, the graph isn’t the whole argument. The algebra shows why each feature appears. Comparison of asymptote and intercept rules for the two AHL rational-function forms.

Rational functions work well with both approaches. A visual method quickly reveals branches and end behaviour; an analytic method gives exact equations for asymptotes and intercepts. The two representations should check each other, not compete.