Question

The G.C.D. and L.C.M. of two polynomials is \[\left( {x - 1} \right)\] and their L.C.M. is \[{x^6} - 1\] . If one of the polynomials is \[{x^3} - 1\] , then the other polynomial is _____________
A \[{x^3} - 1\]
B \[{x^4} - {x^3} + x - 1\]
C \[{x^2} - x + 1\]
D \[{x^2} - 1\]

Answer 1

Hint: The greatest common divisor of two integers, also known as GCD, is the greatest positive integer that divides the two integers. In the given question, we have been given GCD (which is \[\left( {x - 1} \right)\] and LCM (least common multiple, which is \[{x^6} - 1\] ) of two polynomials (out of which one is \[{x^3} - 1\] ). We have to find the other unknown polynomial. This question can be solved by using the relationship between the GCD, LCM and the product of two numbers (or two polynomials). So, we need to just substitute in the values of given quantities and find the unknown polynomial.
 \[GCD \times LCM\] = product of the numbers

Complete step-by-step answer:
Given,
G.C.D. = \[\left( {x - 1} \right)\]
L.C.M. = \[{x^6} - 1\]
First polynomial = \[{x^3} - 1\]
Hence, we need to calculate the second polynomial i.e., y
 \[GCD \times LCM\] = product of the numbers
Substitute the values in the formula as:
 \[\left( {x - 1} \right) \times \left( {{x^6} - 1} \right) = \left( {{x^3} - 1} \right)y\]
We know that,
 \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Hence, we get:
 \[\dfrac{{\left( {x - 1} \right)\left( {{x^3} - 1} \right)\left( {{x^3} + 1} \right)}}{{\left( {{x^3} - 1} \right)}} = y\]
Solving the expression, we have:
 \[y = \left( {x - 1} \right)\left( {{x^3} + 1} \right)\]
or, \[y = {x^4} - {x^3} + x - 1\]
Therefore, the other polynomial is \[{x^4} - {x^3} + x - 1\] .
So, the correct answer is “Option B”.

Note: To solve this question, we must know all the basic elementary functions, like addition, subtracting, squaring the terms, multiplication etc. Here, we have to note that just substitution of formula with respect to GCD and LCM we need to find the other polynomial with respect to the given polynomial. Product of two polynomials says that the product of the GCD and LCM of two numbers (or two polynomials) is equal to their product.