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).

8th Grade Math Practice

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