Solving Binomials by Factoring

Introduction: 

In algebra, the polynomials which have two terms are called binomials. To factor binomials, we need to follow the following methods:

(i) 2a + ab = a(2 + b ) [ Here the given expression has two terms, where a is the common value]    
     = a( 2 + b)         
(ii) a² – b² = ( a + b) ( a – b)   [ This is the standard form]
(iii) (a + b)² = ( a + b)(a +b)
(iv) (a - b)² = ( a - b)(a - b)

Product of two polynomials will give three terms.
( a + b)² = ) a² +b² + 2ab
( a - b)² = ) a² +b² - 2ab

Let us see few problems on this topic solving binomials by factoring.

Example Problems on Solving Binomials by Factoring

Ex 1: Solve (x + 3) (x – 4) = 0

Soln: Given: (x + 3) (x – 4) = 0
This implies: x + 3 = 0 or x – 4 = 0
That is : x = -3, 4.
Therefore the solution is { -3, 4}.

Ex 2: Solve x + y = 7 and xy = 12, find x and y.

Soln: Given : x + y = 7 -----------(1)
                            xy = 12 ---------(2)
Therefore, x – y = sqrt [(x+y)² – 4xy]
                             = sqrt[ 7² – 4(12)]
                             = 1
Therefore, x – y = 1 ---------------(3)
From (1) and (3), we get:
x + y = 7 -----------(1)
x – y = 1 -----------(3)
2x = 8, this implies that x = 4.
Therefore, (1) implies 4 + y = 7
Hence y =3.
Therefore the solution is {4,3}.

Ex 3: Solve x - y = 5 and xy = 24, find the value of x + y.

Soln: x + y = sqrt [(x-y)² + 4xy]
                   = sqrt[ 5² – 4(24)]
                   = 11.
Therefore from, x + y = 11
                             x – y = 5,
We get 2x = 16.
Therefore, x = 8.
Hence from x + y = 11. y = 3.
Therefore the solution is { 8, 3}.

Ex 4: Solve x + y = 11 and xy = 24, find the value of x² – y².

Soln: x – y = sqrt [(x+y)² – 4xy]
                   = sqrt[ 11² – 4(24)]
                   = 5
Therefore, x² – y² = ( x + y )( x – y)
                             = 11 x 5 = 55.

Practice Problems on Solving Binomials by Factoring

1. If a + b = 9  and ab = 36, find a - b
[Ans: a – b = 5]

2. If a – b = 4 and ab = 12, find a² – b².
[Ans: a² – b² = 32]