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Find the least common multiple of x^3-x^2+x-1 and x^2-1. Write the answer in factored form.

The aim of this article is to find the LCM of the two given Polynomial Expressions.

LCM stands for Least Common Multiple, defined as the smallest multiple that is common between the required numbers for which LCM is to be determined. The LCM of two or more polynomial expressions is represented by the expression or factor having the lowest power such that all the given polynomials can be divisible by that factor.

LCM can be found by three methods:

  1. LCM by using factorization
  2. LCM by using repeated division
  3. LCM by using multiple

Following is the Step-by-Step Procedure to calculate the $LCM$ $Least$ $Common$ $Multiple$ of two or more polynomial expressions by using the method of Factorization

(i) Resolve each of the given polynomial expressions into its factors.

(ii) The factors having the highest power, or the highest degree in each expression, will be multiplied to calculate the $LCM$ for the given polynomial expression.

(iii) In the presence of numerical coefficients or constants, calculate their $LCM$ also.

(iv) Multiply the $LCM$ of factors with the highest power and $LCM$ of coefficients or constants to calculate the $LCM$ of given polynomial expressions.

Expert Answer

Given that:

Polynomial Expression # $1$:

\[x^3-x^2+x-1\]

Polynomial Expression # $2$:

\[x^2-1\]

As per the Step-by-Step Procedure to calculate the $LCM$ $Least$ $Common$ $Multiple$ of two or more polynomial expressions by using the method of Factorization, we will first factorize both expressions.

Factorization of Polynomial Expression # $1$:

\[x^3-x^2+x-1\ =\ x^2(x-1)+(x-1)\]

Taking $(x-1) $ common, we get:

\[x^2(x-1)+(x-1)\ =\ {(x}^2+1)(x-1)\]

So, as per calculated above, we have 2 factors for Polynomial Expression # $1$:

\[{(x}^2+1)\ and\ (x-1)\]

Factorization of Polynomial Expression # $2$:

By using the formula for $a^2-b^2\ =\ (a+b)\ (a-b)$, we get:

\[x^2-1\ =\ (x+1)(x-1)\]

So, as per calculated above, we have 2 factors for Polynomial Expression # $2$:

\[(x+1)\ and\ (x-1)\]

Now, to calculate the $LCM$ for the given polynomial expression, the factors having the highest power, or the highest degree in each expression will be multiplied.

Factors for both polynomial expressions are:

\[(x+1)\ ,\ (x-1)\ and\ {(x}^2+1)\]

As all of them have the same power or degree, $Least$ $Common$ $Multiple$ will be calculated by multiplying these factors.

\[Least\ Common\ Multiple\ LCM\ =(x+1)\ (x-1)\ {(x}^2+1)\ \]

Numerical Result

The $Least$ $Common$ $Multiple$ $LCM$ of the polynomial expressions $x^3-x^2+x-1$ and $x^2-1$ in factored form is given below:

\[Least\ Common\ Multiple\ LCM\ =(x+1)\ (x-1)\ {(x}^2+1)\]

Example

Calculate the $LCM$ of given two polynomial expressions: $x^2y^2-x^2$ and $xy^2-2xy-3x$

Solution:

Given that:

Polynomial Expression # $1$:

\[x^2y^2-x^2\]

Polynomial Expression # $2$:

\[xy^2-2xy-3x\]

Factorization of Polynomial Expression # $1$:

\[x^2y^2-x^2\ =\ x^2(\ y^2-1)\]

By using the formula for $a^2-b^2\ =\ (a+b)\ (a-b)$, we get:

\[x^2y^2-x^2\ =\ x^2(y+1)(\ y-1)\]

Factorization of Polynomial Expression # $2$:

\[xy^2-2xy-3x\ =\ x\left(y^2-2y-3\right)\]

\[xy^2-2xy-3x\ =\ x\left(y^2-3y+y-3\right)\]

\[xy^2-2xy-3x\ =\ x[y\left(y-3)+(y-3\right)]\]

\[xy^2-2xy-3x\ =\ x\left(y-3)(y+1\right)\]

Factors with highest power for both polynomial expressions are:

\[x^2\ ,\ (y+1)\ ,\ (\ y-1)\ and\ (\ y-3)\]

$Least$ $Common$ $Multiple$ will be calculated by multiplying these factors.

\[Least\ Common\ Multiple\ LCM\ =\ x^2(y+1)\ (y-1)\ (y-3)\ \]

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