Highest Common Factor of Polynomials
How
to find the highest common factor of polynomials?
To find the highest common factor (H.C.F.) of
polynomials, we first find the factors of polynomials by the method of
factorization and then adopt the same process of finding H.C.F.
Solved
examples to find H.C.F. of polynomials:
1. Find the H.C.F. of 4x
^{2}  9y
^{2} and 2x
^{2} – 3xy.5
Solution:
Factorizing 4x
^{2}  9y
^{2}, we get
(2x)
^{2}  (3y)
^{2}, by using the identities of a
^{2}  b
^{2}.
= (2x + 3y) (2x  3y)
Also, factorizing 2x
^{2} – 3xy by taking the common factor 'x', we get
= x(2x – 3y)
Therefore, H.C.F. of the polynomial 4x
^{2}  9y
^{2} and 2x
^{2} – 3xy is (2x  3y).
2. Find the H.C.F. of the polynomials x
^{2} + 4x + 4 and x
^{2} – 4.
Solution:
Factorizing x
^{2} + 4x + 4 by using the identities (a + b)
^{2}, we get
(x)
^{2} + 2(x)(2) + (2)
^{2}
= (x + 2)
^{2}
= (x + 2) (x + 2)
Also, factorizing x
^{2} – 4, we get
(x)
^{2} – (2)
^{2}, by using the identities of a
^{2}  b
^{2}.
= (x + 2) (x  2)
Therefore, H.C.F. of x
^{2} + 4x + 4 and x
^{2} – 4 is (x + 2).
3. Find the highest common factor of polynomials x
^{2} + 15x + 56, x
^{2} + 5x  24 and x
^{2} + 8x.
Solution:
Factorizing x
^{2} + 15x + 56 by splitting the middle term, we get
(x)
^{2} + 8x + 7x + 56
= x(x + 8) + 7(x + 8)
= (x + 8) (x + 7)
Factorizing x
^{2} + 5x  24, we get
(x)
^{2} + 8x  3x  24
= x(x + 8)  3(x + 8)
= (x + 8) (x  3)
Factorizing x
^{2} + 8x by taking the common factor 'x', we get
= x(x + 8)
Therefore, H.C.F. of x
^{2} + 15x + 56, x
^{2} + 5x  24 and x
^{2} + 8x is (x + 8).
4. Find the H.C.F. x
^{2} – 5x + 4, x
^{2} – 2x + 1 and x
^{2} – 1.
Solution:
Factorizing the quadratic trinomial x
^{2} – 5x + 4, we get
(x)
^{2} – x – 4x + 4
= x(x  1) – 4(x – 1)
= (x  4) (x  1)
Factorizing x
^{2} – 2x + 1 by using the identities (a  b)
^{2}, we get
(x)
^{2} – 2 (x) (1) + (1)
^{2}
= (x – 1)
^{2}
Factorizing x
^{2} – 1 by using the differences of two squares, we get
= x
^{2} – 1
^{2}
= (x + 1) (x – 1)
Therefore, H.C.F. of x
^{2} – 5x + 4, x
^{2} – 2x + 1 and x
^{2} – 1 is (x – 1).
8th Grade Math Practice
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