Question

Find the LCM of the following: \[2{x^3} + 15{x^2} + 2x - 35,{x^3} + 8{x^2} + 4x - 21\] whose GCD is \[x + 7\]
A.\[\left( {2{x^2} + x - 5} \right)\left( {{x^3} + 8{x^2} + 4x - 21} \right)\]
B.\[\left( {2{x^2} - x - 5} \right)\left( {{x^3} + 8{x^2} + 4x - 21} \right)\]
C.\[\left( {2{x^2} + x - 5} \right)\left( {{x^3} - 8{x^2} + 4x - 21} \right)\]
D.None of these

Answer 1

Hint: Here, we will find the LCM of the given polynomials by using the relation between GCD and LCM. Greatest Common Divisor (GCD) is defined as the greatest number which divides exactly both the numbers. Least Common Multiple (LCM) is defined as the smallest number which is divisible by both the numbers.

Complete step-by-step answer:
We are given with two polynomials \[2{x^3} + 15{x^2} + 2x - 35,{x^3} + 8{x^2} + 4x - 21\].
Let \[P\left( x \right) = 2{x^3} + 15{x^2} + 2x - 35\] and \[Q\left( x \right) = {x^3} + 8{x^2} + 4x - 21\] .
We are also given that the GCD of two polynomials is \[x + 7\].
The relation between GCD and LCM is given by the product of their polynomials and the product of their GCD and LCM
\[P\left( x \right) \cdot Q\left( x \right) = GCD\left( {P\left( x \right)\& Q\left( x \right)} \right) \cdot LCM\left( {P\left( x \right)\& Q\left( x \right)} \right)\] where \[P\left( x \right)\] and \[Q\left( x \right)\] are the polynomials respectively.
By rewriting the relation, we get
\[ \Rightarrow LCM\left( {P\left( x \right)\& Q\left( x \right)} \right) = \dfrac{{P\left( x \right) \cdot Q\left( x \right)}}{{GCD\left( {P\left( x \right)\& Q\left( x \right)} \right)}}\]
Now, substituting the given polynomials and GCD of the given polynomials, we get
\[ \Rightarrow LCM\left( {P\left( x \right)\& Q\left( x \right)} \right) = \dfrac{{\left( {2{x^3} + 15{x^2} + 2x - 35} \right) \cdot \left( {{x^3} + 8{x^2} + 4x - 21} \right)}}{{x + 7}}\]
Now, we will divide the polynomial \[P\left( x \right) = 2{x^3} + 15{x^2} + 2x - 35\]by the GCD of the given polynomials by using long division, we get
\[{\rm{ }}x + 7\mathop{\left){\vphantom{1\begin{array}{l}2{x^3} + 15{x^2} + 2x - 35\\\underline {{}^{\left( - \right)}2{x^3}\mathop + \limits^{\left( - \right)} 14{x^2}} \\{\rm{ }}{x^2} + 2x\\\underline {{\rm{ }}{}^{\left( - \right)}{x^2}\mathop + \limits^{\left( - \right)} 7x} \\{\rm{ }} - 5x - 35\\\underline {{\rm{ }}{}^{\left( + \right)} - 5x\mathop - \limits^{\left( + \right)} 35} \\\underline {{\rm{ }}0} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{array}{l}2{x^3} + 15{x^2} + 2x - 35\\\underline {{}^{\left( - \right)}2{x^3}\mathop + \limits^{\left( - \right)} 14{x^2}} \\{\rm{ }}{x^2} + 2x\\\underline {{\rm{ }}{}^{\left( - \right)}{x^2}\mathop + \limits^{\left( - \right)} 7x} \\{\rm{ }} - 5x - 35\\\underline {{\rm{ }}{}^{\left( + \right)} - 5x\mathop - \limits^{\left( + \right)} 35} \\\underline {{\rm{ }}0} \end{array}}}}
\limits^{\displaystyle\,\,\, {2{x^2} + x - 5}}\]
Thus, we get the quotient as \[2{x^2} + x - 5\] .
\[ \Rightarrow LCM\left( {P\left( x \right)\& Q\left( x \right)} \right) = \left( {2{x^2} + x - 5} \right) \cdot \left( {{x^3} + 8{x^2} + 4x - 21} \right)\]
Therefore, the LCM of the given polynomials \[P\left( x \right)\& Q\left( x \right)\]is \[\left( {2{x^2} + x - 5} \right) \cdot \left( {{x^3} + 8{x^2} + 4x - 21} \right)\].
Thus Option (A) is the correct answer.

Note: We should note that when dividing the polynomial, it is enough to divide either one of the two polynomials by the GCD of the polynomials. We might make a mistake by dividing both the polynomials by the GCD of the polynomials. The polynomial which is not divided by the GCD of the polynomial remains the same as it is. If the LCM of the two polynomials is known, then the same relation can be used to find the GCD of the two polynomials.