Relationship Between LCM and HCF of Two Polynomials
The product of polynomials equals the product of its H.C.F. and L.C.M., which is the common relationship between L.C.M. and H.C.F. of polynomials. The following is one way to express this relationship.
If p(x) and q(x) are two polynomials, then
p(x) ∙ q(x) = {H.C.F. of p(x) and q(x)} x {L.C.M. of p(x) and q(x)}
Example: Find the HCF of p(x) = 3xy and q(x) = 2x2 if LCM of p(x) and q(x) is 6x2y.
Solution:
As we know, HCF (a, b) × LCM (a, b) = a × b
⇒ HCF × 6x2y = 3xy × 2x2
⇒ HCF = 3xy × 2x2 / 6x2y
⇒ HCF = x
Thus, HCF of p(x) and q(x) is x.
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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