Question

Find the LCM of the following: $2x^2-18y^2,5x^{2}y+15x{y}^2,x^3+27y^3$

• A. $10xy(x+3y)(x-3y)(x^2-3xy+9y^2)$
• B. $10xy(x-3y)(x-3y)(x^2-3xy+9y^2)$
• C. $10xy(x+3y)(x+3y)(x^2-3xy+9y^2)$
• D. None of these

Answer 1

The correct option is A $10xy(x+3y)(x-3y)(x^2-3xy+9y^2)$

We know that the least common multiple (LCM) is the smallest number or expression that is a common multiple of two or more numbers or algebraic terms.

We first factorize the given binomials $2x^2-18y^2,5x^{2}y+15x{y}^2$ and $x^3+27y^3$ as shown below:

$2x^2-18y^2=2(x^2-9y^2)=2(x^2-(3y{)}^2)=2(x-3y\left)\right(x+3y\left)\right(\because a^2-b^2=(a-b)(a+b))$

$5x^{2}y+15x{y}^2=5xy(x+3y)$

$x^3+27y^3=x^3+(3y{)}^3=(x+3y\left)\right(x^2+(3y{)}^2-(x\times 3y\left)\right)(\because a^3+b^3=(a+b\left)\right(a^2+b^2-ab\left)\right)=(x+3y)(x^2+9y^2-3xy)$

Now multiply all the factors, using each common factor only once, therefore, the LCM is:

$2\times 5xy\times (x-3y)\times (x+3y)\times (x^2+9y^2-3xy)=10xy(x-3y)(x+3y)(x^2+9y^2-3xy)$

Hence, the LCM of $2x^2-18y^2,5x^{2}y+15x{y}^2$ and $x^3+27y^3$ is $10xy(x-3y)(x+3y)(x^2+9y^2-3xy)$.