C4 – Algebra and Functions Summary
 Remainder Theorem: When the polynomial f(x) is divided by (ax  b)
b
.
a
C4 – Algebra and Functions Summary
 Remainder Theorem: When the polynomial f(x) is divided by (ax  b)
b
.
a
the remainder = f 
the remainder = f 
b
 = 0.
a
b
 = 0.
a
 Factor Theorem: If (ax – b) is a factor of f(x) then f 
 Factor Theorem: If (ax – b) is a factor of f(x) then f 
 The degree of a polynomial is the highest power of x it contains.
 In improper algebraic fraction the degree of the numerator is greater
than or equal to the degree of the denominator.
 To simplify an improper algebraic fraction use division. If the numerator
has degree n and the denominator has degree m (n > m) then the
quotient has degree n – m and the remainder is a proper fraction.
 A rational function is an algebraic fraction.
 The degree of a polynomial is the highest power of x it contains.
 In improper algebraic fraction the degree of the numerator is greater
than or equal to the degree of the denominator.
 To simplify an improper algebraic fraction use division. If the numerator
has degree n and the denominator has degree m (n > m) then the
quotient has degree n – m and the remainder is a proper fraction.
 A rational function is an algebraic fraction.
Partial Fractions
 Means writing an algebraic fraction as a sum of several fractions.
 If a fraction in improper always simplify first and express the remainder
as partial fractions.
Partial Fractions
 Means writing an algebraic fraction as a sum of several fractions.
 If a fraction in improper always simplify first and express the remainder
as partial fractions.
 Denominator has linear factors:
 Denominator has linear factors:
px  q
A
B
.


(ax  b)(cx  d ) ax  b cx  d
px  q
A
B
.


(ax  b)(cx  d ) ax  b cx  d
To solve, multiply everything by (ax + b)(cx + d).
 Denominator has repeated factor:
To solve, multiply everything by (ax + b)(cx + d).
 Denominator has repeated factor:
To solve, multiply everything by (ax + b)(cx + d)2.
 You may need to factorise the denominator first before proceeding.
 To find the numerators of the partial fractions after you multiplied up
i) expand brackets and simplify. The coefficients of each power of x
must be the same on both sides of the equation. OR
ii) substitute values in for x to create simultaneous equations. Choosing
x such that the value of a bracket becomes zero makes things
considerably simpler.
To solve, multiply everything by (ax + b)(cx + d)2.
 You may need to factorise the denominator first before proceeding.
 To find the numerators of the partial fractions after you multiplied up
i) expand brackets and simplify. The coefficients of each power of x
must be the same on both sides of the equation. OR
ii) substitute values in for x to create simultaneous equations. Choosing
x such that the value of a bracket becomes zero makes things
considerably simpler.
px  q
A
B
C



2
ax  b cx  d (cx  d ) 2
(ax  b)(cx  d )
px  q
A
B
C



2
ax  b cx  d (cx  d ) 2
(ax  b)(cx  d )