FACTORING POLYNOMIALS USING ALGEBRAIC IDENTITIES

Trinomials—Look for Perfect-Square Trinomials :

a2 + 2ab + b2  =  (a + b)2

a2 - 2ab + b2  =  (a - b)2

Examples : 

x2 + 4x + 4  =  x2 + 2(x)(2) + 22  =  (x + 2)2

m2 - 6m + 9  =  m2 - 2(m)(3) + 32  =  (m - 3)2

Binomials—Look for a Difference of Two Squares :

a2 - b2  =  (a + b)(a - b)

Example : 

4x2 - 9y2  =  22x2 - 32y2

=  (2x)2 - (3y)2

=  (2x + 3y)(2x - 3y)

Solved Problems

Problem 1 :

Factor :

x2 + 6xy + 9y2

Solution : 

x2 + 6xy + 9y=  x2 + 2(x)(3y) + (3y)2

=  (x + 3y)2

=  (x + 3y)(x + 3y)

Problem 2 :

Factor :

4a2 - 20ab + 25b2

Solution : 

4a2 - 20ab + 25b2  =  22a2 - 20ab + 52b2

=  (2a)2 - 20ab + (5b)2

=  (2a)2 - 2(2a)(5b) + (5b)2

=  (2a - 5b)2

=  (2a - 5b)(2a - 5b)

Problem 3 :

Factor :

2x2 + 12xy + 18y2

Solution : 

Factor out the GCF.

=  2(x2 + 6xy + 9y2)

=  2[x2 + 2(x)(3y) + (3y)2]

=  2(x + 3y)2

=  2(x + 3y)(x + 3y) 

Problem 4 :

Factor :

2ab2 - 16ab + 32a

Solution : 

Factor out the GCF.

=  2a(b2 - 8b + 16)

=  2a[b2 - 2(b)(4) + 42]

=  2a(b - 4)2

Problem 5 :

Factor :

x2 - 25y2

Solution : 

x2 - 25y2  =  x2 - 52y2

=  x2 - (5y2)

=  (x + 5y)(x - 5y)

Problem 6 :

Factor :

9m2 - 16n2

Solution : 

9m2 - 16y2  =  32m2 - 42n2

=  (3m)2 - (4n2)

=  (3m + 4n)(3m - 4n)

Problem 7 :

Factor :

x4 - y4

Solution : 

x4 - y4  =  (x2)2 - (y2)2

=  (x2 + y2)(x2 - y2)

=  (x2 + y2)(x + y)(x - y)

Problem 8 :

Factor :

8p4 - 18q4

Solution : 

Factor out the GCF.

32p4 - 162q4  =  2(16p4 - 81q4)

=  2[42(p2)2 - 92(q2)2]

=  2[(4p2)2 - (9q2)2]

=  2(4p2 + 9q2)(4p2 - 9q2)

=  2(4p2 + 9q2)[(2p)2 - (3q)2]

=  2(4p2 + 9q2)(2p + 3q)(2p - 3q)

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