No matter how many polynomials are given, you can easily find the great common factor of those polynomials by finding the factors. Find the factors of numerical and literal coefficients. Multiply the common factors out of all polynomials to get the highest common factor. Check out some example questions and answers on HCF of Polynomials by Factorization.
Solved Examples on G.C.F of Polynomials
Example 1.
Find the Greatest Common Factor of x² – 2x + 2, x⁴ – 1, x³ – 2x² – 5x + 6?
Solution:
Factorizing x² – 2x + 2 by using (a – b)².
= (x – 1)²
= (x – 1) ( x – 1)
Factorizing x⁴ – 1 by using a² – b².
= (x²)² – 1²
= (x² – 1) (x² + 1)
= ((x)² – 1²) (x² + 1)
= (x + 1) (x – 1) (x² + 1)
Factorizing x³ – 2x² – 5x + 6 by splitting the middle terms
= x³ – x² – x² – 6x + x + 6
= x²(x – 1) – x (x – 1) – 6 (x – 1)
= (x – 1) (x² – x – 6)
= (x – 1) (x² – 3x + 2x – 6)
= (x – 1) (x(x – 3) + 2 (x – 3))
= (x – 1) (x – 3) (x + 2)
The common factor of x² – 2x + 2, x⁴ – 1, x³ – 2x² – 5x + 6 is (x – 1)
Therefore, H.C.F of x² – 2x + 2, x⁴ – 1, x³ – 2x² – 5x + 6 is (x – 1).
Example 2.
Calculate the Highest Common Factor of x²y² – x² and xy² – 2xy – 3x by factorization.
Solution:
Factorizing x²y² – x² by taking the x² common.
= x² (y²- 1)
= x² (y² – 1²)
= x² ( y + 1) ( y – 1)
= x * x ( y + 1) ( y – 1)
Factorizing xy² – 2xy – 3x by taking the variable x common.
= x (y² – 2y – 3)
= x (y² – 3y + y – 3)
= x (y(y – 3) + 1( y – 3))
= x (y – 3) ( y + 1)
The common factors of x²y² – x² and xy² – 2xy – 3x are x , (y – 1).
Therefore, the H.C.F of x²y² – x² and xy² – 2xy – 3x is x * (y – 1).
Example 3.
Find the H.C.F of x⁴ – y⁴ and x²(x – y) + y²(x – y) + y – x.
Solution:
Factorizing x⁴ – y⁴ by using a² – b² formula.
= (x²)² – (y²)²
= (x² + y²) (x² – y²)
= (x² + y²) (x + y) ( x – y)
Factorizing x²(x – y) + y²(x – y) + y – x by taking (x – y) common
= (x – y) (x² + y² – 1)
The common factor of x⁴ – y⁴ and x²(x – y) + y²(x – y) + y – x are (x – y)
Therefore, the H.C.F of x⁴ – y⁴ and x²(x – y) + y²(x – y) + y – x is (x – y).
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