Teacher2Teacher Q&A #18119

Teachers' Lounge Discussion: "Bottoms up" as a method to factor trinomials

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From: Ranjan Biswas

To: Teacher2Teacher Public Discussion
Date: 2007073106:58:54
Subject: Factorisation of trinomials

I teach factorisation in little bit different way. If ax^2 + bx + c is a given trinomial. Step 1: Multiply a and c Step 2: Check the sign of c Step 3: a)If c is positive, I tell them to break ac into two factor whose sum is the middle term b and the sum must be within a bracket, no matter what is the sign of the middle term b. b)If c is negative, I tell them to break ac into two factors whose difference is the middle term b and the difference must be within bracket, no matter what is the sign of the middle term b. Step 3: Open the bracket. Step 4: Take out common term from the first two term and take out common term from the last two term. Check that the two brackets after taking common are same. Step 5: The common bracket we take common to write the trinomial as factors. EXAMPLE1 2X^2 + 5X + 3 Step 1 multiply 2 and 3 and we get 6 Step 2 The last sign is positive ( + ) Step 3: a) The last sign is positive ( + ) Hence we have to break down 6 into two factors whose sum is 5 3 and 2 are the two factors as their sum is 5. We now write the polynomial as 2x^2 + ( 3x + 2x ) + 3 Open the bracket we get 2x^2 + 3x + 2x + 3 Step 4: 2x^2 + 3x + 2x + 3 = x(2x + 3) + 1(2x + 3) = (2x + 3)(x + 1) EXAMPLE2 Now let's see for the polynomial 2x^2 - 5x + 3 Step 1 multiply 2 and 3 and we get 6 Step 2 The last sign is positive ( + ) Step 3: a) The last sign is positive ( + ) Hence we have to break down 6 into two factors whose sum is 5 ignoring the sign of the middle term. 3 and 2 are the two factors as their sum is 5. We now write the polynomial as 2x^2 - ( 3x + 2x ) + 3 Open the bracket we get 2x^2 - 3x - 2x + 3 Step 4: 2x^2 - 3x - 2x + 3 = x(2x - 3) - 1(2x - 3) = (2x - 3)(x - 1) EXAMPLE3 Now for the polynomial 2x^2 + x -3 Step 1 multiply 2 and 3 and we get 6 Step 2 The last sign is negative ( - ) Step 3: a) The last sign is negative ( - ) Hence we have to break down 6 into two factors whose difference is 1 3 and 2 are the two factors as their difference is 1. We now write the polynomial as 2x^2 + ( 3x - 2x ) + 3 Open the bracket we get 2x^2 + 3x - 2x - 3 Step 4: 2x^2 + 3x - 2x - 3 = x(2x + 3) - 1(2x + 3) = (2x + 3)(x - 1) EXAMPLE4 Now for the polynomial 2x^2 - x -3 Step 1 multiply 2 and 3 and we get 6 Step 2 The last sign is negative ( - ) Step 3: a) The last sign is negative ( - ) Hence we have to break down 6 into two factors whose difference is 1 3 and 2 are the two factors as their difference is 1. We now write the polynomial as 2x^2 - ( 3x - 2x ) + 3 Open the bracket we get 2x^2 - 3x + 2x - 3 Step 4: 2x^2 - 3x + 2x - 3 = x(2x - 3) + 1(2x - 3) = (2x - 3)(x + 1) In this method I found that the students are not confused about the signs to be taken to form the middle term.

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