Factorization of algebraic expressions

PROGRAMME F2

INTRODUCTION TO ALGEBRA

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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

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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

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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 

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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 )

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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:

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(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

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(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.

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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:

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.

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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

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Worked examples and exercises are in the text