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 How To Factor An Expression Factoring an algebraic expression is essential in mathematics. In arithmetic, you learn the fundamental theorem of arithmetic, it's about factoring composite numbers. In algebra, you learn the fundamental theorem of algebra, it's about factoring reducible polynomials. Here you will learn the basic techniques in factoring algebraic expressions. (A)  Factoring a monomial from a polynomial. Factor out (if there is one), the greatest common  factor of all terms in the expression Ex 1:  Factor       Ex 2 :   Factor .     Ex 3:  Factor   Ex 4 :   Factor (B) Factoring binomials of the following types. (i)  Using the formula to factor the difference of two squares: Ex 1:  Factor Ex 2 : Factor . Ex 3 :    Factor .           Ex 4 :   Factor . (ii)  Using the formula to factor the sum or the difference of two cubes:                   Ex 1:  Factor          Ex 2:   Factor                     Ex 3:   Factor           Ex 4: : Factor .          Ex 5:   Factor             (C)  Factoring second degree trinomials: (i) Perfect square of a trinomial: (ii) Trinomial of the form . (i) Using the formula: Ex 1: Factor                    Ex 2 :   Factor  Ex 3 :   Factor            (ii)      Second degree trinomial of the form Case 1:     when a = 1,       Ex 1:   Factor     We look for two whole numbers whose product is 6 and their sum is 5 Decomposing 6 as product of two factors we get:      The desired factors are 2 and 3. so Ex 2 : Factor        The desired factors are -16 and 3. so   Ex 3:   Factor  None of the factors added up to be a sum of -7 and so the trinomial can't be factored, it is an irreducible polynomial. Case 2: when a is not 1: If the trinomial can be broken in to factor then , where p,q are factors of a and r,s are factors of c and bx is the sum of psx+qrx. Ex 1:Factor Factoring by grouping we get Ex 2:Factor Factoring by grouping we get Ex 3: Factor If factor then Where p,q can be 15, 1 or 5, 3 and m,n can be 8, 1 or 4, 2 So