Solved Examples of HCF and LCM of Two Polynomials
Example 1: Find the H.C.F. and L.C.M. of the expressions x2 β 5x + 6 and x2 β 7x + 10 by factorization.
Solution:
x2 β 5x + 6 = x2 β 2x β 3x +6
β x2 β 5x + 6 = 2(x-2) -3(x-2)
β x2 β 5x + 6 = (x β 2) (x β 3)x2 β 7x + 10 = x2 β 2x -5x + 10
β x2 β 7x + 10 = x(x-2) -5(x-2)
β x2 β 7x + 10 = (x-2) (x-5)LCM = (x-2) Γ (x-3) Γ (x-5)
HCF = (x-2)
Example 2: Find LCM and HCF of polynomials (x+3) (6x2 + 5x -4) and (2x2 + 7x + 3) (x + 3).
Solution:
(x+3) (6x2 + 5x -4) = (x+3) (6x2 + 8x -3x -4)
β (x+3) (6x2 + 5x -4) = (x+3) [2x(3x+4) -1 (3x+4)]
β (x+3) (6x2 + 5x -4) = (x+3) (2x -1) (3x+4)(2x2 + 7x +3) (x+3) = (2x2 + 6x + x+3) (x+3)
β (2x2 + 7x +3) (x+3) = [2x(x+3) +1 (x+3)] (x+3)
β (2x2 + 7x +3) (x+3) = (2x+1) (x+3) (x+3)
β (2x2 + 7x +3) (x+3) = (2x+1) (x+3)2LCM = (x+3)2 Γ (2x-1) Γ(3x+4) Γ(2x+1)
HCF = (x+3)
Example 3: Find HCF and LCM of (x2 + xy + y2) and (x3 β y3 )
Solution:
- x2 + xy + y2
- x3 β y3= (x-y) (x2+ xy + y2)
HCF = x2 + xy + y2
LCM = (x2 + xy + y2) (x-y) = x3 β y3
Example 4: Find HCF and LCM of x2-9 and x2 β 6x + 9.
Solution:
x2-9 = (x2) β (3)2 = (x-3) (x+3)
x2β 6x + 9 = x2 β 3x -3x +9
β x2β 6x + 9 = x(x-3) -3(x-3)
β x2β 6x + 9 = (x-3) (x-3) or (x-3)2HCF = (x-3)
LCM = (x-3)2 (x+3)
Example 5: Determine the HCF and LCM of the polynomials 4a2 b, 6ab and 8ab2.
Solution:
- 4a2 b = 22
- 6ab = 2 x 3
- 8ab2 = 23
HCF = 2 Γ a Γ b = 2ab
LCM = 23 Γ 3 Γ a2 Γ b2 = 24a2 b2
HCF and LCM of Polynomials
HCF (Highest Common Factor) and LCM (Least Common Multiple) of polynomials are concepts similar to those for integers. The HCF of two polynomials is the largest polynomial that divides both polynomials without leaving a remainder, while the LCM is the smallest polynomial that is a multiple of both polynomials.
To find the HCF of polynomials, we take the common factors among all the factors of two polynomials, and for LCM, we take the product of all their unique factors. In this article, we will discuss how to find HCF and LCM for polynomials, with some solved examples as well.
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