Factorization by Using Identities
Factorization by using identities will help us to factorize
an algebraic expression easily.
The following
identities are:
(i) (a + b)
^{2} = a
^{2} + 2ab +b
^{2},
(ii) (a - b)
^{2} = a
^{2} - 2ab + b
^{2} and
(iii) a
^{2} – b
^{2} = (a + b)(a – b).
Now we will use these identities to factorize the given algebraic expressions.
Solved
examples on factorization by using identities:
1. Factorize using
the formula of square of the sum of two terms:
(i) z^{2} + 6z + 9
Solution:
We can express z
^{2} + 6z + 9 as using a
^{2} + 2ab + b
^{2} = (a + b)
^{2}
= (z)
^{2} + 2(z)(3) + (3)
^{2}
= (z + 3)
^{2}
= (z + 3)(z + 3)
(ii) x
^{2} + 10x + 25
Solution:
We can express x
^{2} + 10x + 25 as using a
^{2} + 2ab + b
^{2} = (a + b)
^{2}
= (x)
^{2} + 2 ( x)( 5) + (5)
^{2}
= (x + 5)
^{2}
= (x + 5)(x - 5)
2. Factorize using the formula of square of the difference of two terms:
(i) 4m
^{2} – 12mn + 9n
^{2}
Solution:
We can express 4m
^{2} – 12mn + 9n
^{2} as using a
^{2} - 2ab + b
^{2} = (a - b)
^{2}
= (2m)
^{2} - 2(2m)(3n) + (3n)
^{2}
= (2m – 3n)
^{2}
= (2m - 3n)(2m - 3n)
(ii) x
^{2} - 20x + 100
Solution:
We can express x
^{2} - 20x + 100 as using a
^{2} - 2ab + b
^{2} = (a - b)
^{2}
= (x)
^{2} - 2(x)(10) + (10)
^{2}
= (x - 10)
^{2}
=(x - 10)(x - 10)
3. Factorize using the formula of difference of two squares:
(i) 25x
^{2} - 49
Solution:
We can express 25x
^{2} - 49 as using a
^{2} – b
^{2} = (a + b)(a - b).
= (5x)
^{2} - (7)
^{2}
= (5x + 7)(5x - 7)
(ii) 16x
^{2} – 36y
^{2}
Solution:
We can express 16x
^{2} – 36y
^{2} as using a
^{2} – b
^{2} = (a + b)(a - b).
= (4x)
^{2} - (6y)
^{2}
= (4x + 6y)(4x – 6y)
(iii) 1 – 25(2a – 5b)
^{2}
Solution:
We can express 1 – 25(2a – 5b)
^{2} as using a
^{2} – b
^{2} = (a + b)(a - b).
= (1)
^{2} - [5(2a – 5b)]
^{2}
= [1 + 5(2a – 5b)] [1 - 5(2a – 5b)]
= (1 + 10a – 25b) (1 – 10a + 25b)
4. Factor completely using the formula of difference of two squares: m
^{4} – n
^{4}
Solution:
m
^{4} – n
^{4}
We can express m
^{4} – n
^{4} as using a
^{2} – b
^{2} = (a + b)(a - b).
= (m
^{2})
^{2} - (n
^{2})
^{2}
= (m
^{2} + n
^{2})( m
^{2} - n
^{2})
Now again, we can express m
^{2} – n
^{2} as using a
^{2} – b
^{2} = (a + b)(a - b).
= (m
^{2} + n
^{2}) (m + n) (m - n)
8th Grade Math Practice
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