Learn about the relation between the highest common factor and least common factor of two polynomials on this page. First of all, know the definitions of H.C.F, L.C.M, and the relationship between them. Check the solved example questions in the below sections. We have listed the step-by-step solutions for all the example problems provided making it easy for you to understand the concepts.
H.C.F and L.C.M Definition
The highest common factor (H.C.F) is defined as the largest term that divides evenly into all the terms from a group.
Example: H.C.F of 12 and 15 is 3. Because multiples of 12 are 2, 2, 3, and multiples of 15 are 5, 3.
The Lowest Common Multiple (L.C.M) is defined as the smallest term that is a multiple of all numbers from a group.
Example: L.C.M of 12, 15 is 60. Because multiples of 12 are 2, 2, 3, multiples of 15 are 3, 5. And the L.C.M is the multiplication of the smallest common number and remaining numbers from the group.
Relation Between H.C.F. and L.C.M. of Two Polynomials
The relation between L.C.M and H.C.F of polynomials is the product of polynomials is equal to the product of its H.C.F and L.C.M. This relationship can be expressed as follows.
p(x) * q(x) = {L.C.M of p(x) and q(x)} * {H.C.F of p(x) and q(x)]}.
So, find the highest common factor, least common factor of polynomials by using the factorization method. And multiply them to find the product of two polynomial expressions. Using this formula, you can easily find one polynomial when L.C.M, H.C.F of both polynomials, and other polynomial is given.
Solved Examples on Relationship Between H.C.F. and L.C.M. of Two Polynomials
Example 1.
Find H.C.F and L.C.M of two polynomials 2x² – x – 1 and 4x² + 8x + 3 and prove that the product of polynomials is the product of their L.C.M and H.C.F?
Solution:
Given two polynomials are 2x² – x – 1 and 4x² + 8x + 3.
By factoring 2x² – x – 1, we get
= 2x² – 2x + x -1
= 2x(x – 1) + 1(x – 1)
= (x – 1) ( 2x + 1)
By factoring 4x² + 8x + 3, we get
= 4x² + 6x + 2x + 3
= 2x (2x + 3) + 1(2x + 3)
= (2x + 3) (2x + 1)
The common factor in both polynomials is (2x + 1).
Therefore, H.C.F = common factor = (2x + 1)
L.C.M = common factor * remaining factors
= (2x + 1) * (2x + 3) (x – 1) = (2x + 1) (2x + 3) (x – 1)
The relation between H.C.F. and L.C.M. of Two Polynomials is Product of Two Polynomials = Product of Polynomials L.C.M and H.C.F
(2x² – x – 1) (4x² + 8x + 3) = (2x + 1) (2x + 3) (x – 1) (2x + 1)
(x – 1) ( 2x + 1) (2x + 3) (2x + 1) = (2x + 1)²(2x + 3) (x – 1)
Hence proved.
Example 2.
Find the G.C.F of polynomials x³ + y³ and x⁴ + x²y² + y⁴ whose L.C.M is (x³ + y³) (x² + xy + y²)?
Solution:
Given two polynomials are x³ + y³ and x⁴ + x²y² + y⁴
L.C.M of polynomials is (x³ + y³) (x² + xy + y²)
By factoring x⁴ + x²y² + y⁴, we get
= (x² + y²)² – (xy)²
= (x² + xy + y²)² (x² – xy + y²)²
Product of Two Polynomials = Product of Polynomials L.C.M and H.C.F
(x³ + y³) (x⁴ + x²y² + y⁴) = (x³ + y³) (x² + xy + y²) * H.C.F
Cancel the common term (x³ + y³)
(x⁴ + x²y² + y⁴) = (x² + xy + y²) * H.C.F
(x² + xy + y²)² (x² – xy + y²)² = (x² + xy + y²) * H.C.F
Therefore, H.C.F = (x² – xy + y²)²
Example 3.
L.C.M and G.C.F of two polynomials are x³ – 10x² + 11x + 70 and x – 7. And one of the polynomial is x² – 12x + 35, find the other polynomial?
Solution:
Given that,
L.C.M, H.C.F of two polynomials is x³ – 10x² + 11x + 70, x – 7
First polynomial = x² – 12x + 35
Product of Two Polynomials = Product of Polynomials L.C.M and H.C.F
x² – 12x + 35 * Second polynomial = (x³ – 10x² + 11x + 70) * (x – 7)
Second polynomial = (x³ – 10x² + 11x + 70) * (x – 7) / (x² – 12x + 35)
So, second polynomial = (x + 2) (x – 7)
= x² – 7x + 2x -14 = x² -5x – 14.