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Q&A #18119

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

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From: Ranjan Biswas <RanjanB@kis.in>
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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