Find the LCM of the following

10(9x² + 6xy + y²), 12(3x² – 5xy – 2y²), 14(6x⁴ + 2x³)

Given terms: –

10(9x² + 6xy + y²), 12(3x² – 5xy – 2y²), 14(6x⁴ + 2x³)

Formula used: –

LCM = Least Common Multiple

Means it is the lowest term by which every element must be

divided completely;

10(9x² + 6xy + y²) = 2 × 5 × ((3x)² + 2 × 3x × y + y²)

= 2 × 5 × (3x + y)²

= 2 × 5 × (3x + y) × (3x + y)

12(3x² – 5xy – 2y²) = 2 × 2 × 3 × (3x² – 6xy + xy – 2y²)

= 2 × 2 × 3 × (3x(x – 2y) + y(x – 2y))

= 2 × 2 × 3 × (x – 2y) × (3x + y)

14(6x⁴ + 2x³) = 2 × 7 × 2 × x × x × x × (3x + 1)

⇒ first find the common factors in all terms

Common factor = 2

Common factors in any 2 terms

2 × [(5(3x + 4)²)(2 × 3 × (x – 2y)(3x + y))(7 × 2 × x³ × (3x + 1))]

2 × 2 × (3x + y)[(5(3x + 4))(3 × (x – 2y))(7 × x³ × (3x + 1))]

⇒ then multiply the remaining factors of terms in common

factor to get the LCM

= 2 × 2 × 5 × 3 × 7 × x³ × (3x + y)(3x + y)(x – 2y)(3x + 1)

= 420x³(3x + y)²(x – 2y)(3x + 1)

Conclusion: –

The LCM of given terms [10(9x² + 6xy + y²), 12(3x² – 5xy – 2y²), 14(6x⁴ + 2x³)] is 420x³(3x + y)²(x – 2y)(3x + 1)