Algebra: partial fractions, series, binomial expansion

Pure Mathematics 3 (P3) – Cambridge 9709 – Core Algebra & Calculus

1. Algebra

1.1 Partial‑Fraction Decomposition

For a rational function \(\displaystyle \frac{P(x)}{Q(x)}\) with \(\deg P\le\deg Q\) first factorise the denominator \(Q(x)\) completely over the real numbers. The form of the decomposition depends on the type of factor that occurs.

Factor of \(Q(x)\)	Term(s) in the decomposition
\((ax+b)\) (linear)	\(\displaystyle \frac{A}{ax+b}\)
\((ax+b)^k\) (repeated linear)	\(\displaystyle \frac{A_1}{ax+b}+ \frac{A_2}{(ax+b)^2}+ \dots+ \frac{A_k}{(ax+b)^k}\)
\(ax^{2}+bx+c\) (irreducible quadratic)	\(\displaystyle \frac{Bx+C}{ax^{2}+bx+c}\)

If \(\deg P\ge\deg Q\) perform polynomial long division first, then decompose the resulting proper fraction.

Example (including a repeated linear factor)

\[ \frac{2x^{3}+3x^{2}+5x+4}{(x-1)(x^{2}+1)^{2}} =\frac{A}{x-1}+ \frac{Bx+C}{x^{2}+1}+ \frac{Dx+E}{(x^{2}+1)^{2}} . \] Multiplying by the denominator and equating coefficients gives \(A=1,\;B=2,\;C=-1,\;D=0,\;E=3\). Hence \[ \frac{2x^{3}+3x^{2}+5x+4}{(x-1)(x^{2}+1)^{2}} =\frac{1}{x-1}+ \frac{2x-1}{x^{2}+1}+ \frac{3}{(x^{2}+1)^{2}} . \]

1.2 Logarithmic & Exponential Functions

• Laws of logarithms (any base) \[ \log(ab)=\log a+\log b,\qquad \log\!\left(\frac{a}{b}\right)=\log a-\log b,\qquad \log(a^{k})=k\log a . \]
• Basic exponential rules \[ a^{b+c}=a^{b}a^{c},\qquad (a^{b})^{c}=a^{bc},\qquad a^{-b}= \frac{1}{a^{b}} . \]
• Solving equations – isolate the exponential or logarithm and apply the above laws.
  Example: Solve \(3^{2x-1}=5\). \[ 2x-1=\frac{\ln5}{\ln3}\;\Longrightarrow\; x=\frac12\Bigl(1+\frac{\ln5}{\ln3}\Bigr). \]
• Inequalities – \(\log x\) (for \(x>0\)) and \(e^{x}\) are strictly increasing, so the direction of the inequality is preserved when applying them.
• Linearisation of data – to test whether a relationship is exponential or a power law, plot
  • \(\ln y\) versus \(x\) (exponential) or
  • \(\log y\) versus \(\log x\) (power law)
  A straight line indicates the appropriate model.

1.3 Trigonometry

All identities listed in the Cambridge syllabus are required.

• Reciprocal functions: \(\displaystyle \sec x=\frac1{\cos x},\; \csc x=\frac1{\sin x},\; \cot x=\frac1{\tan x}\).
• Pythagorean identities: \[ \sin^{2}x+\cos^{2}x=1,\qquad 1+\tan^{2}x=\sec^{2}x,\qquad 1+\cot^{2}x=\csc^{2}x . \]
• Compound‑angle formulas: \[ \sin(A\pm B)=\sin A\cos B\pm\cos A\sin B, \] \[ \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B, \] \[ \tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}. \]
• Double‑angle and half‑angle: \[ \sin2x=2\sin x\cos x,\qquad \cos2x=\cos^{2}x-\sin^{2}x=2\cos^{2}x-1=1-2\sin^{2}x, \] \[ \tan\frac{x}{2}= \frac{1-\cos x}{\sin x}= \frac{\sin x}{1+\cos x}. \]
• Solving trigonometric equations – reduce to a single function using the identities above, then apply the general solutions \[ \sin\theta =k\;\Longrightarrow\;\theta =(-1)^{n}\arcsin k+n\pi, \] \[ \cos\theta =k\;\Longrightarrow\;\theta =\pm\arccos k+2n\pi, \qquad \tan\theta =k\;\Longrightarrow\;\theta =\arctan k+n\pi, \] where \(n\in\mathbb Z\).

1.4 Binomial Expansion

• Integer exponent \(n\ge0\) (standard binomial theorem) \[ (a+b)^{n}= \sum_{r=0}^{n}\binom{n}{r}a^{\,n-r}b^{\,r}, \qquad \binom{n}{r}= \frac{n!}{r!\,(n-r)!}. \]
• General (fractional or negative) exponent – the binomial series (valid for \(|b/a|<1\)): \[ (a+b)^{n}=a^{n}\Bigl[1+n\frac{b}{a}+\frac{n(n-1)}{2!}\Bigl(\frac{b}{a}\Bigr)^{2} +\frac{n(n-1)(n-2)}{3!}\Bigl(\frac{b}{a}\Bigr)^{3}+\cdots\Bigr]. \]
• Example – Expand \((1+x)^{\frac12}\) up to the term in \(x^{3}\): \[ (1+x)^{\frac12}=1+\frac12x-\frac18x^{2}+\frac1{16}x^{3}+O(x^{4}). \]

1.5 Series and Summation

• Arithmetic series – sum of the first \(n\) terms: \[ S_{n}= \frac{n}{2}\bigl(a_{1}+a_{n}\bigr)=\frac{n}{2}\bigl(2a_{1}+(n-1)d\bigr). \]
• Geometric series \[ S_{n}=a\frac{1-r^{n}}{1-r}\quad(req1),\qquad S_{\infty}= \frac{a}{1-r}\;( |r|<1). \]
• Convergence tests (syllabus level)
  • Ratio test – for \(\displaystyle \sum a_{k}\), if \(\displaystyle L=\lim_{k\to\infty}\bigl|\frac{a_{k+1}}{a_{k}}\bigr|<1\) the series converges; if \(L>1\) it diverges.
  • p‑test – \(\displaystyle \sum\frac{1}{k^{p}}\) converges for \(p>1\) and diverges for \(p\le1\).
• Maclaurin/Taylor series (up to required order) \[ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{f^{(3)}(0)}{3!}x^{3}+\cdots . \] Useful standard expansions: \[ e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots, \qquad \sin x =x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\cdots, \qquad \cos x =1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\cdots . \]
• Example – Approximate \(\ln(1.2)\) using the first three non‑zero terms of the Maclaurin series for \(\ln(1+x)\): \[ \ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots, \] with \(x=0.2\): \[ \ln(1.2)\approx0.2-\frac{0.2^{2}}{2}+\frac{0.2^{3}}{3}=0.1827\;( \text{to 4 d.p.} ). \]

2. Calculus – Differentiation, Integration & Applications

2.1 Advanced Differentiation

Function	Derivative
\(e^{ax}\)	\(ae^{ax}\)
\(\ln(ax+b)\)	\(\displaystyle \frac{a}{ax+b}\)
\(\sin(ax+b)\)	\(a\cos(ax+b)\)
\(\cos(ax+b)\)	\(-a\sin(ax+b)\)
\(\tan(ax+b)\)	\(a\sec^{2}(ax+b)\)
\(\displaystyle \frac{u}{v}\) (quotient rule)	\(\displaystyle \frac{u'v-uv'}{v^{2}}\)
\(u\cdot v\) (product rule)	\(u'v+uv'\)
\(y\) defined implicitly by \(F(x,y)=0\)	\(\displaystyle \frac{dy}{dx}= -\frac{F_{x}}{F_{y}}\)
\(y\) defined parametrically \((x=t,\;y=f(t))\)	\(\displaystyle \frac{dy}{dx}= \frac{dy/dt}{dx/dt}\)

Example – Implicit differentiation for the ellipse \(\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\):

\[ \frac{2x}{a^{2}}+\frac{2y}{b^{2}}\frac{dy}{dx}=0 \;\Longrightarrow\; \frac{dy}{dx}= -\frac{b^{2}x}{a^{2}y}. \]

2.2 Advanced Integration

Integrand	Integral \((+C)\)
\(e^{ax}\)	\(\displaystyle \frac{1}{a}e^{ax}\)
\(\sin(ax+b)\)	\(-\frac{1}{a}\cos(ax+b)\)
\(\cos(ax+b)\)	\(\displaystyle \frac{1}{a}\sin(ax+b)\)
\(\sec^{2}(ax+b)\)	\(\displaystyle \frac{1}{a}\tan(ax+b)\)
\(\dfrac{1}{ax+b}\)	\(\displaystyle \frac{1}{a}\ln|ax+b|\)
\((ax+b)^{n},\;neq-1\)	\(\displaystyle \frac{(ax+b)^{n+1}}{a(n+1)}\)
\(\displaystyle \frac{u'}{u}\)	\(\ln|u|\)

Key techniques required by the syllabus

1. Substitution (u‑substitution) – let \(u\) be a function whose derivative appears in the integrand.
2. Integration by parts – \(\displaystyle \int u\,dv = uv-\int v\,du\).
3. Partial‑fraction integration – first decompose the rational function (see 1.1) then integrate each term using the table above.
4. Trigonometric substitution – used for integrands containing \(\sqrt{a^{2}-x^{2}},\sqrt{a^{2}+x^{2}}\) or \(\sqrt{x^{2}-a^{2}}\):
   • \(x=a\sin\theta\) for \(\sqrt{a^{2}-x^{2}}\)
   • \(x=a\tan\theta\) for \(\sqrt{a^{2}+x^{2}}\)
   • \(x=a\sec\theta\) for \(\sqrt{x^{2}-a^{2}}\)

Example – Integrate \(\displaystyle \frac{x}{x^{2}+1}\,dx\) (substitution)

\[ u=x^{2}+1\;\Longrightarrow\;du=2x\,dx\;\Longrightarrow\; \int\frac{x}{x^{2}+1}\,dx=\frac12\int\frac{du}{u} =\frac12\ln|u|+C=\frac12\ln(x^{2}+1)+C. \]

Example – Trigonometric substitution for \(\displaystyle \int\frac{dx}{\sqrt{a^{2}-x^{2}}}\)

\[ x=a\sin\theta,\;dx=a\cos\theta\,d\theta,\; \sqrt{a^{2}-x^{2}}=a\cos\theta, \] \[ \int\frac{dx}{\sqrt{a^{2}-x^{2}}} =\int\frac{a\cos\theta\,d\theta}{a\cos\theta} =\int d\theta=\theta+C =\arcsin\!\left(\frac{x}{a}\right)+C . \]

2.3 First‑Order Linear Differential Equations

Standard form: \(\displaystyle \frac{dy}{dx}+P(x)y=Q(x)\).

1. Find the integrating factor \(\displaystyle \mu(x)=e^{\int P(x)\,dx}\).
2. Multiply the whole equation by \(\mu(x)\); the left‑hand side becomes \(\displaystyle \frac{d}{dx}\bigl[\mu(x)y\bigr]\).
3. Integrate and solve for \(y\).

Example – Solve \(\displaystyle \frac{dy}{dx}+2y=4e^{-x}\).

\[ \mu(x)=e^{\int2dx}=e^{2x}, \qquad \frac{d}{dx}\bigl(e^{2x}y\bigr)=4e^{x}. \] Integrating, \[ e^{2x}y=4e^{x}+C\;\Longrightarrow\; y=4e^{-x}+Ce^{-2x}. \]

3. Summary of Core Content

• Partial‑fraction decomposition (linear, repeated linear, irreducible quadratic).
• Logarithmic and exponential laws, solving equations/inequalities, data linearisation.
• All required trigonometric identities and solution methods.
• Binomial expansion for integer and non‑integer exponents.
• Series: arithmetic, geometric, convergence tests, Maclaurin/Taylor expansions.
• Differentiation: basic functions, product/quotient, implicit and parametric.
• Integration: basic forms, substitution, parts, partial fractions, trig substitution.
• First‑order linear differential equations.

These notes are written to match the Cambridge 9709 (Pure Mathematics 3) syllabus for the 2026‑27 academic year. Material marked as “extended” (e.g., change‑of‑base) is optional and may be omitted in a strict syllabus‑only revision.