There are several methods that can be used to solve polynomials equations. The primary goal of Factoring Quadratic Polynomials is to express a quadratic equation as a product of multiplication. The common methods include; Box Method, Diamond Method, square root method, and factorizing by grouping. Quadratic equations have the first factor with a degree (n) of ‘2’. The general formula of a quadratic equation is P(X) = ax n + bx +c. where a, b, and c are coefficients which are always integers and n=2. The nature of the equation sometimes dictate the method used. For example square root method is used where we have one coefficient in most cases ‘a’. Factorizing by GCF can only be applied where we have a common factor between a, b, and c (Pearson, n.d.) . This paper focuses on factoring trinomials using the group method.
Grouping method
The method is used where we have all coefficients given like in 2x 2 -3x -9 = 0, a=2, b=-3, c= -9. The quadratic expression could be solved using grouping technique and distributive property. The first step in factoring is to find two integers say r and s whose sum is b(r+s=-3) and whose product is ac(r*s=-18) (Patrick, 2010)
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Step 1 : Find two integers
| R*s=-18 | R+s |
| 1*-18=18 | 1+(-18)=-17 |
| 2*-9=-18 | 2+(-9)=-7 |
| 3*-6=-18 | 3+(-6)=-3 |
The two integers are 3 and -6 where r=3 s=-6
Step 2 : Rewrite the trinomial as ax 2 +rx-sx- c
2x 2 -6x +3x -9
The -3x has been replaced with -6x +3x
Step 3 : consider the terms in pairs
(2x2-6x) + (3x-9)
Find common factor using distributive property. The common factor in first group is 2x while the second group is 3
2x(x-3) +3(x-3)
Step 4 : Factored equation:
(2x+3) (x- 3)
Expanding the expression
2x2+ 3x-6x-9
2x 2 -3x-9
Example two: 2x 2 +11x+ 15 a=2, b=11, c=15
The first step in factoring is to find two integers say r and s whose sum is b(r+s=11) and whose product is ac(r*s=30)
Step 1 Find two integers
| R*s=-18 | R+s |
| 1*30=30 | 1+30=31 |
| 2*15=30 | 2+15=17 |
| 3*10=30 | 3+10=13 |
| 5*6=30 | 5+6=11 |
The two integers are 5 and 6 where r=6 s=-5
Step 2 : Rewrite the trinomial as ax2+rx-sx- c
2x2 +6x +5x +15
The 11x has been replaced with 6x +5x
Step 3 : consider the terms in pairs
(2x2+6x) +(5x+ 15)
Find common factor using distributive property. The common factor in first group is 2x while the second group is 5
2x(x+3) +5(x+3)
Step 4 : Factored Equation
(2x+5) (x +3)
Expanding the expression
2x2 +5x +6x +15
2x2 +11x +15
References
Patrick, J. (2010, April 25). Factoring Trinomials: Factor by Grouping - ex 1 . Retrieved from You Tube: https://www.youtube.com/watch?v=HvBiJ9W00Z4
