Factorize by Grouping The Terms
Factorize by
grouping the terms (two or more) means that we need to group the terms which
have common factors before factoring.
Method to factorize by grouping the
terms:
(i) From the groups of the given expression a common factor
can be taken out from each group.
(ii) Factorize each group
(iii) Now take out the factor common to group formed.
Now we will learn how to factorize by grouping two or more terms.
Solved
examples to factorize by
grouping the terms:
1. Factorize
grouping the following expressions:
(i) 18a
^{3}b
^{3} - 27a
^{2}b3 + 36a3b
^{2}
Solution:
18a
^{3}b
^{3} - 27a
^{2}b3 + 36a3b
^{2}
= 9a
^{2}b
^{2}(2ab – 3b + 4a)
(ii) 12x
^{2}y
^{3} - 21x
^{3}y
^{2}
Solution:
12x
^{2}y
^{3} - 21x
^{3}y
^{2}
= 3x
^{2}y
^{2}(4y - 7x)
(iii) y
^{3} - y
^{2} + y - 1
Solution:
y
^{3} - y
^{2} + y - 1
= y
^{2}(y - 1) + 1(y - 1)
= (y - 1) (y
^{2} + 1)
(iv) axy + bcxy – az – bcz
Solution:
axy + bcxy – az – bcz
= xy(a + bc) – z(a + bc)
= (a + bc) (xy – z)
(v) x
^{2} - 3x – xy + 3y
Solution:
x
^{2} - 3x – xy + 3y
= x(x - 3) - y(x - 3)
= (x - 3) (x – y)
2. How to factorize by grouping the following expressions?
(i) 2x
^{4} – x
^{3} + 4x - 2
Solution:
2x
^{4} – x
^{3} + 4x - 2
= x
^{3}(2x – 1) + 2(2x - 1)
= (2x – 1) (x
^{3} + 2)
(ii) pr + qr - ps - qs
Solution:
pr + qr - ps - qs
= r(p + q) - s(p + q)
= (p + q) (r - s)
(iii) mx - my - nx - ny
Solution:
mx - my - nx - ny
= m(x - y) - n(x - y)
= (x - y) (m - n)
3. How to
factorize by grouping the algebraic expressions?
(i) a
^{2}c
^{2} + acd + abc + bd
Solution:
a
^{2}c
^{2} + acd + abc + bd
= ac(ac + d) + b(ac + d)
= (ac + d) (ac + b)
(ii) 5a + ab + 5b + b
^{2}
Solution:
5a + ab + 5b + b
^{2}
= a(5 + b) + b(5 + b)
= (5 + b) (a + b)
(iii) ab - by - ay + y
^{2}
Solution:
ab - by - ay + y
^{2}
= b(a - y) - y(a - y)
= (a - y) (b - y)
4.
Factorize the expressions:
(i) x
^{4} + x
^{3} + 2x + 2
Solution:
x
^{4} + x
^{3} + 2x + 2
= x
^{3}(x + 1) + 2 (x + 1)
= (x + 1) (x
^{3} + 2)
(ii) f
^{2}x
^{2} + g
^{2}x
^{2} – ag
^{2} – af
^{2}
Solution:
f
^{2}x
^{2} + g
^{2}x
^{2} – ag
^{2} – af
^{2}
= x
^{2}(f
^{2} + g
^{2}) – a (g
^{2} + f
^{2})
= x
^{2}(f
^{2} + g
^{2}) – a(f
^{2} + g
^{2})
= (f
^{2} + g
^{2})(x
^{2} – a)
5. Factorize by grouping the terms (a
^{2} + 3a)
^{2} - 2(a
^{2} + 3a) – b(a
^{2} + 3a) + 2b
Solution:
(a
^{2} + 3a)
^{2} - 2(a
^{2} + 3a) – b(a
^{2} + 3a) + 2b
= [(a
^{2} + 3a)
^{2} - 2(a
^{2} + 3a)] – [b(a
^{2} + 3a) - 2b]
= (a
^{2} + 3a) (a
^{2} + 3a - 2) – b(a
^{2} + 3a - 2)
= (a
^{2} + 3a - 2) (a
^{2} + 3a - b)
8th Grade Math Practice
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