Algebra

Need to know

What you need to know

• First, ensure the fraction is 'proper' (degree of numerator < degree of denominator). If not, perform polynomial division first.
• Fully factorise the denominator to identify the types of factors.
• Write down the partial fraction decomposition with unknown constants (A, B, C, etc.). Use $Ax+B$ for irreducible quadratic factors.
• Solve for the constants by either substituting convenient values of $x$ or by equating coefficients of powers of $x$.

Explanation

The P3 Algebra Toolkit

1. P3 modulus: |f(x)| graphs reflect negative portions of the y-values above the x-axis. For a linear function like $|ax+b|$, this creates a 'V' shape with the vertex on the x-axis.
2. Partial fractions break down complex fractions. For an irreducible quadratic denominator like $x^2+4$, the numerator must be a linear term, $Ax+B$. For a repeated factor like $(x-c)^2$, you need separate fractions for $(x-c)$ and $(x-c)^2$.
3. The binomial expansion works for rational powers 'n', like in $(1+x)^n$. The result is an infinite series that is only a valid approximation when the magnitude of the 'x' term is less than 1.
4. For a polynomial $P(x)$, if $P(a)=0$, then $(x-a)$ is a factor. If the polynomial has real coefficients and a complex number $z = a+bi$ is a root, its conjugate $z^* = a-bi$ must also be a root.