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Learn to solve the given algebraic expressions using the below formulas.
(i) a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b)
(ii) a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b)
Factoring Perfect Square Trinomials Examples
1. Factorization when the given expression is a perfect square
(i) m4 – 10m2n2 + 25n4
Solution:
Given expression is m4 – 10m2n2 + 25n4
The given expression m4 – 10m2n2 + 25n4 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = m2, b = 5n2
Apply the formula and substitute the a and b values.
m4 – 10m2n2 + 25n4
(m2)2 – 2 (m2) (5n2) + (5n2)2
(m2 – 5n2)2
(m2 – 5n2) (m2 – 5n2)
Factors of the m4 – 10m2n2 + 25n4 are (m2 – 5n2) (m2 – 5n2)
(ii) b2+ 6b + 9
Solution:
Given expression is b2+ 6b + 9
The given expression b2+ 6b + 9 is in the form a2 + 2ab + b2.
So find the factors of given expression using a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b) where a = b, b = 3
Apply the formula and substitute the a and b values.
b2+ 6b + 9
(b)2 + 2 (b) (3) + (3)2
(b + 3)2
(b + 3) (b + 3)
Factors of the b2+ 6b + 9 are (b + 3) (b + 3)
(iii) p4 – 2p2 q2 + q4
Solution:
Given expression is p4 – 2p2 q2 + q4
The given expression p4 – 2p2 q2 + q4 is in the form a2 – 2ab + b2.
So find the factors of given expression using a2 – 2ab + b2 = (a – b)2 = (a – b) (a – b) where a = p2, b = q2
Apply the formula and substitute the a and b values.
p4 – 2p2 q2 + q4
(p2)2 – 2 (p2) (q2) + (q2)2
(p2 – q2)2
(p2 – q2) (p2 – q2)
From the formula (a2 – b2) = (a + b) (a – b), rewrite the above equation.
(p + q) (p – q) (p + q) (p – q)
Factors of the p4 – 2p2 q2 + q4 are (p + q) (p – q) (p + q) (p – q)
2. Factor using the identity
(i) 25 – a2 – 2ab – b2
Solution:
Given expression is 25 – a2 – 2ab – b2
Rearrange the given expression as 25 – (a2 + 2ab + b2)
a2 + 2ab + b2 = (a + b)2 = (a + b) (a + b)
25 – (a + b)2
(5)2– (a + b)2
From the formula (a2 – b2) = (a + b) (a – b), rewrite the above equation.
[(5 + a + b)(5 – a – b)]
(ii) 1- 2mn – (m2 + n2)
Solution:
Given expression is 1- 2mn – (m2 + n2)
1- 2mn – m2 – n2
1 – (2mn + m2 + n2)
1 – (m + n)2
(1)2 – (m + n)2
From the formula (a2 – b2) = (a + b) (a – b), rewrite the above equation.
(1 + m + n) (1 – m + n)