STROUD Worked examples and exercises are in the text PROGRAMME F2 INTRODUCTION TO ALGEBRA

STROUD Worked examples and exercises are in the text Algebraic expressions Powers Logarithms Multiplication of algebraic expressions of a single variable Division of one expression by another Factorization of algebraic expressions Programme F2: Introduction to algebra

STROUD Worked examples and exercises are in the text 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 Programme F2: Introduction to algebra

STROUD Worked examples and exercises are in the text Factorization of algebraic expressions Common factors Programme F2: Introduction to algebra The simplest form of factorization is the extraction of highest common factors from a pair of expressions. For example:

STROUD Worked examples and exercises are in the text Factorization of algebraic expressions Common factors by grouping Programme F2: Introduction to algebra Four termed expressions can sometimes be factorized by grouping into two binomial expressions and extracting common factors from each. For example:

STROUD Worked examples and exercises are in the text Factorization of algebraic expressions Useful products of two simple factors Programme F2: Introduction to algebra A number of standard results are worth remembering:

STROUD Worked examples and exercises are in the text Factorization of algebraic expressions Quadratic expressions as the product of two simple factors Programme F2: Introduction to algebra

STROUD Worked examples and exercises are in the text Factorization of algebraic expressions Factorization of a quadratic expression ax 2 + bx + c when a = 1 Programme F2: Introduction to algebra The factorization is given as: Where, if c > 0, f 1 and f 2 are factors of c whose sum equals b, both factors having the same sign as b. If c < 0, f 1 and f 2 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 ax 2 + bx + c when a ≠ 1 [clarified by J.A.B.] [OPTIONAL] The factorization can be found by first re-expressing as follows: Where, if c > 0, f 1 and f 2 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: f 1 and f 2 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 Factorization of algebraic expressions Test for simple factors Programme F2: Introduction to algebra The quadratic expression: Has simple factors if, and only if: