CIE A Level Maths: Pure 3复习笔记1.3.1 Linear Denominators

What are partial fractions?

• This is the reverse process to adding (or subtracting) fractions
• When adding fractions a common denominator is required
• In partial fractions the common denominator is split into parts (factors)
• Partial fractions are used in binomial expansions (see Multiple GBEs) and integration (see Integration by Parts)

What are linear denominators?

• A linear factor is of the form (ax + b)
• A non-linear denominator may be written as the product of linear factors
• If the denominator can be factorised

How do I find partial fractions?

STEP 1        Factorise the polynomial in the denominator

(Sometimes the numerator can be factorised too)

STEP 2        Split the fraction into a sum with single linear denominators

STEP 3        Multiply by the denominator to get rid of fractions

STEP 4        Substitute values of x to find A, B, etc

(An alternative method is comparing coefficients)

STEP 5        Write the original as partial fractions

Comparing coefficients

• The quantity of each term must be equal on both sides
• “The number of x² on the LHS” = “The number of x² on the RHS”
• “The number of …” is called the coefficient of x²