Question

Find the LCM of the polynomials $ {x^2} + 9x + 18 $ and $ x + 3 $ .

Answer 1

Hint: LCM is the least common multiple, it is also termed as lowest common multiple and least common divisor. LCM is that smallest positive integer which can be divided by all the given numbers. Let us take an example, LCM of $ 3 $ and $ 12 $ is $ 12 $ as it is the least common multiple and $ 12 $ is that number which can be divided by $ 3 $ as well as with $ 12 $ .

Complete step by step solution:
To find the LCM of the polynomial, we need to factorize the given polynomial and we will consider both the factors as the numbers. Now, we will factorize the given polynomial $ {x^2} + 9x + 18 $ , this is a quadratic equation and we know that the general form of the quadratic equation $ a{x^2} + bx + c $ and to find the factors we need to multiply a and c and then we have to find the numbers that on multiplying gives the product of $ ac $ and on adding or subtracting it results to b. So, for this equation $ {x^2} + 9x + 18 $ , $ 3 $ and $ 6 $ are the numbers which on multiplying gives $ 18 $ and on adding gives $ 9 $ . Now, we can write the equation as,
 $
   \Rightarrow {x^2} + 3x + 6x + 18 \\
   \Rightarrow x\left( {x + 3} \right) + 6\left( {x + 3} \right) \\
   \Rightarrow \left( {x + 6} \right)\left( {x + 3} \right) \;
  $
Hence, these are the factors of the given polynomial, now we have $ \left( {x + 6} \right)\left( {x + 3} \right) $ and $ \left( {x + 3} \right) $ and $ \left( {x + 3} \right) $ is common in both polynomials and now, we have left with $ \left( {x + 6} \right) $ . So, the LCM of the polynomials are $ \left( {x + 3} \right)\left( {x + 6} \right) = {x^2} + 9x + 18 $ .
So, the correct answer is “$ \left( {x + 3} \right)\left( {x + 6} \right) = {x^2} + 9x + 18 $”.

Note: Polynomials are the algebraic expressions which form by variables and coefficients. Let us understand the polynomial with the given expression in the question i.e, $ {x^2} + 9x + 18 $ , in this expression there are three terms $ {x^2} $ , $ 9x $ and $ 18 $ , $ {x^2} $ is the variable and its coefficient is $ 1 $ and x is the variable whose coefficient is 9.