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 4x2 - 9y2 and 2x2 – 3xy.5

Solution:

Factorizing 4x2 - 9y2, we get

(2x)2 - (3y)2, by using the identities of a2 - b2.

= (2x + 3y) (2x - 3y)

Also, factorizing 2x2 – 3xy by taking the common factor 'x', we get

= x(2x – 3y)

Therefore, H.C.F. of the polynomial 4x2 - 9y2 and 2x2 – 3xy is (2x - 3y).



2. Find the H.C.F. of the polynomials x2 + 4x + 4 and x2 – 4.

Solution:

Factorizing x2 + 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 x2 – 4, we get

(x)2 – (2)2, by using the identities of a2 - b2.

= (x + 2) (x - 2)

Therefore, H.C.F. of x2 + 4x + 4 and x2 – 4 is (x + 2).



3. Find the highest common factor of polynomials x2 + 15x + 56, x2 + 5x - 24 and x2 + 8x.

Solution:

Factorizing x2 + 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 x2 + 5x - 24, we get

(x)2 + 8x - 3x - 24

= x(x + 8) - 3(x + 8)

= (x + 8) (x - 3)

Factorizing x2 + 8x by taking the common factor 'x', we get

= x(x + 8)

Therefore, H.C.F. of x2 + 15x + 56, x2 + 5x - 24 and x2 + 8x is (x + 8).



4. Find the H.C.F. x2 – 5x + 4, x2 – 2x + 1 and x2 – 1.

Solution:

Factorizing the quadratic trinomial x2 – 5x + 4, we get

(x)2 – x – 4x + 4

= x(x - 1) – 4(x – 1)

= (x - 4) (x - 1)

Factorizing x2 – 2x + 1 by using the identities (a - b)2, we get

(x)2 – 2 (x) (1) + (1)2

= (x – 1)2

Factorizing x2 – 1 by using the differences of two squares, we get

= x2 – 12

= (x + 1) (x – 1)

Therefore, H.C.F. of x2 – 5x + 4, x2 – 2x + 1 and x2 – 1 is (x – 1).







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