Question

The L.C.M. of \[{{x}^{3}}-{{y}^{3}},x-y\] and \[{{x}^{2}}+xy+{{y}^{2}}\] will be.
(a) \[{{x}^{2}}-{{y}^{2}}\]
(b) \[x-y\]
(c) \[{{x}^{3}}-{{y}^{3}}\]
(d) \[x\left( x+y \right)\]

Answer 1

Hint: From the definition of polynomials and types of polynomials we have the special products. By using the special products and then calculating the L.C.M. gives the result.
\[{{x}^{3}}-{{a}^{3}}={{\left( x-a \right)}^{3}}+3xa\left( x-a \right)\]

Complete step-by-step answer:

POLYNOMIAL: An expression of the form \[{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+......+{{a}_{n-1}}x+{{a}_{n}},\], where \[{{a}_{0}},{{a}_{1}},{{a}_{2}}.......,{{a}_{n-1}},{{a}_{n}}\] are real numbers and n is a non-negative integer, is called a polynomial in the variable x. Polynomials in variable x are generally denoted by \[f\left( x \right)\].

FUNDAMENTAL OPERATIONS ON POLYNOMIAL

Addition of Polynomials: To calculate the addition of two or more polynomials, we collect different groups of like powers together and add the coefficients of like terms.

Subtraction of Polynomials: To find the subtraction of two or more polynomials, we collect different groups of like powers together and subtract the coefficients of like terms.

Multiplication of Polynomials: Two polynomials can be multiplied by applying distributive law and simplifying the like terms.

SPECIAL PRODUCTS:

\[\begin{align}
  & \left( x-a \right)\left( {{x}^{2}}+ax+{{a}^{2}} \right)={{x}^{3}}-{{a}^{3}} \\
 & {{x}^{3}}-{{a}^{3}}={{\left( x-a \right)}^{3}}+3xa\left( x-a \right) \\
\end{align}\]

L.C.M. OF MONOMIALS: To find the L.C.M. of two monomials, we multiply the L.C.M. of the numerical coefficient of the monomials by all the factors raised to the highest power of each of the letters common to both the polynomials.

Similarly it is done for the polynomials also.

\[\Rightarrow {{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+yx+{{y}^{2}} \right)\]

L.C.M. of \[{{x}^{3}}-{{y}^{3}},x-y\] and \[{{x}^{2}}+xy+{{y}^{2}}\] is given by:

\[\begin{align}
  & \Rightarrow \left( x-y \right)\left( {{x}^{2}}+yx+{{y}^{2}} \right) \\
 & \Rightarrow {{x}^{3}}-{{y}^{3}} \\
\end{align}\]

Hence, the correct option is (c).

Note: It is important to note that as we are calculating the L.C.M. which is also called the least common multiple. We need to multiply the common terms we have for the given three terms because neglecting any one of them does not imply the least common multiple.
Instead of calculating the least common multiple directly we can also do it by first calculating the highest common factor for the given polynomials and then divide the product of the polynomials with it.

\[L.C.M. of polynomials = \dfrac{\text{Product of polynomials}}{\text{H}\text{.C}\text{.F of polynomials}}\]