Transcript Factorization of algebraic expressions

PROGRAMME F2
INTRODUCTION
TO
ALGEBRA
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Algebraic expressions
Powers
Logarithms
Multiplication of algebraic expressions of a single variable
Division of one expression by another
Factorization of algebraic expressions
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Common factors
Common factors by grouping
Useful products of two simple factors
Quadratic expressions as the product of two simple factors
Factorization of a quadratic expression
Test for simple factors
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Common factors
The simplest form of factorization is the extraction of highest common
factors from a pair of expressions. For example:
35x2 y2 10xy3  5xy2  7 x  2 y 
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Common factors by grouping
Four termed expressions can sometimes be factorized by grouping into two
binomial expressions and extracting common factors from each. For
example:
2ac  6bc  ad  3bd
 (2ac  6bc)  (ad  3bd )
 2c(a  3b)  d (a  3b)
 (a  3b)(2c  d )
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Useful products of two simple factors
A number of standard results are worth remembering:
STROUD
(a )
(a  b)2  a2  2ab  b2
(b)
(a  b)2  a 2  2ab  b2
(c)
(a  b)(a  b)  a 2  b2
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Quadratic expressions as the product of two simple factors
STROUD
(a )
( x  g )( x  k )  x2  ( g  k ) x  gk
(b)
( x  g )( x  k )  x2  ( g  k ) x  gk
(c)
( x  g )( x  k )  x2  ( g  k ) x  gk
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Factorization of a quadratic expression ax2 + bx + c when a = 1
The factorization is given as:
x2  bx  c  ( x  f1)(x  f 2 )
Where, if c > 0, f1 and f2 are factors of c whose sum equals b, both factors
having the same sign as b.
If c < 0, f1 and f2 are factors of c with opposite signs, the numerically larger
having the same sign as b and their difference being equal to b.
STROUD
Worked examples and exercises are in the text
Factorization of a quadratic expression ax2 + bx + c when a ≠ 1 [clarified by J.A.B.]
[OPTIONAL]
The factorization can be found by first re-expressing as follows:
ax2  bx  c  ax2  f1x  f 2 x  c
Where, if c > 0, f1 and f2 are two factors of |ac| whose sum equals |b|, both
factors having the same sign as b.
If c < 0 their values differ by the value of |b|, the numerically larger of the
two having the same sign as b and the other factor having the opposite sign.
Note: f1 and f2 do NOT themselves form part of the final factorization.
Then, we use the grouping method for a four-term expression, ending up
with a factorization of form (px +/- g)(qx +/-k). See examples in textbook.
BUT there is an easier, more automatic method that we’ll see once we deal
with the solution of quadratic equations.
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Factorization of algebraic expressions
Test for simple factors
The quadratic expression:
ax2  bx  c
Has simple factors if, and only if:
b2  4ac  k 2 for some integer k
STROUD
Worked examples and exercises are in the text