Question

Find the LCM of $2{{x}^{3}}-3{{x}^{2}}-9x+5$, $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$whose GCD is $2x-1$.

Answer 1

Hint: First, before proceeding for this, we must know the terminology used here which is GCD(greatest common divisor) which actually means highest common factor(HCF). Then, we also know that the multiplication of the two polynomials always gives the product of Cm and HCF. Then, by using the factorization method to get factors of the polynomial $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$, we get the final result.

Complete step-by-step answer:
In this question, we are supposed to find the LCM of $2{{x}^{3}}-3{{x}^{2}}-9x+5$, $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$whose GCD is $2x-1$.
So, before proceeding for this, we must know the terminology used here which is GCD(greatest common divisor) which actually means highest common factor(HCF).
Now, we need to find the LCM of above two polynomials which is least common multiple means it has the multiplication of the common terms with all the remaining terms form the two polynomials.
Then, we also know that the multiplication of the two polynomials always gives the product of Cm and HCF.
So, by using this property, we can calculate the value of LCM of given polynomials as:
$\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( 2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8 \right)=HCF\times LCM$
Then, by substituting the value of HCF as given in the question, we get:
$\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( 2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8 \right)=\left( 2x-1 \right)\times LCM$
Now, by rearranging and solving the above expression we can get the value of LCM as:
$LCM=\dfrac{\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( 2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8 \right)}{\left( 2x-1 \right)}$
Now, by using the factorization method to get factors of the polynomial $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$as:
$2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8=2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-16x+5x+8$
Then, by taking the common terms from the above polynomial, we get:
$\begin{align}
  & 2x\left( {{x}^{3}}-5x-8 \right)-1\left( {{x}^{3}}-5x-8 \right) \\
 & \Rightarrow \left( 2x-1 \right)\left( {{x}^{3}}-5x-8 \right) \\
\end{align}$
Then, by substituting the value of $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$as $\left( 2x-1 \right)\left( {{x}^{3}}-5x-8 \right)$in the LCM expression, we get:
\[LCM=\dfrac{\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( 2x-1 \right)\centerdot \left( {{x}^{3}}-5x-8 \right)}{\left( 2x-1 \right)}\]
Now, by cancelling the like terms from the numerator and denominator, we get the LCM as:
\[LCM=\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( {{x}^{3}}-5x-8 \right)\]
So, we get the LCM of the polynomials $2{{x}^{3}}-3{{x}^{2}}-9x+5$, $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$whose GCD is $2x-1$as \[\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( {{x}^{3}}-5x-8 \right)\].
Hence, \[\left( 2{{x}^{3}}-3{{x}^{2}}-9x+5 \right)\centerdot \left( {{x}^{3}}-5x-8 \right)\]is the correct answer.

Note: Now, to solve these type of the questions we can also use the long division method to find the polynomial quotient when $2{{x}^{4}}-{{x}^{3}}-10{{x}^{2}}-11x+8$is divided by $2x-1$. Then, it will also give the same result as calculated above but it is a little time consuming approach.