How to find the highest common factor of polynomials by factorization?
Let us follow the following examples to know how to find the highest common factor (H.C.F.) or greatest common factor (G.C.F.) of polynomials by factorization.
Solved examples of highest common factor of polynomials by factorization:
1. Find out the H.C.F. of a^{2}b + ab^{2} and a^{2}c + abc by factorization.= ab(a + b)
= a × b × (a + b)
= ac(a + b)
= a × c × (a + b)
It can be seen, in both the expressions ‘a’ and ‘(a + b)’ are the common factors and there is no other common factor.
Therefore, the required H.C.F. a^{2}b + ab^{2} and a^{2}c + abc is a(a + b)= a × a × (b + c)
Second expression = (ab + ac)^{2}= (ab + ac) (ab + ac)
= a(b + c) a(b + c)
= a × a × (b + c) × (b + c)
It can be seen that, in both the expressions ‘a’, ‘a’ and ‘(b + c)’ are the common factors and there is no other common factor.
Therefore, the required H.C.F. is a × a × (b + c) = a^{2}(b + c).= c × (a + b) × (a + b)
Second expression = (a^{2}c^{2}  b^{2}c^{2})= c × c × (a + b) × (a  b)
Third expression = a(ac^{2} + bc^{2})= a × c × c × (a + b)
It can be seen that, c and (a + b) are the common factors of the expressions.
Therefore, the required H.C.F. of c(a + b)^{2}, (a^{2}c^{2} – b^{2}c^{2}) and a(ac^{2} + bc^{2}) is c(a + b)= 3 × x × x × (y + z) × (y + z)
Second expression = 6x(y^{2}  z^{2})= 6x(y + z) (y  z)
= 2 × 3 × x × (y + z) × (y  z)
Therefore, the required H.C.F. is 3 × x × (y + z) = 3x(y + z)
8th Grade Math Practice
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