Do you need any help to find the least common multiple of polynomials? Then read this article to find the L.C.M of polynomials by factorization method. All you need to do is get the factors of each polynomial, multiply the common and remaining terms to get the L.C.M. You can also check the example questions in the below sections.
LCM of Polynomials Solved Examples
Example 1.
Find the lowest common multiple of (2x² – 4x), (3x⁴ – 12x²), and (2x⁵ – 2x⁴ – 4x³)?
Solution:
Given polynomials are (2x² – 4x), (3x⁴ – 12x²), and (2x⁵ – 2x⁴ – 4x³).
First Polynomial = (2x² – 4x)
= 2x(x – 2), by taking 2x common
Second Polynomial = (3x⁴ – 12x²)
= 3x² (x² – 4), by taking 2x² common.
= 3x²(x² – 2²), by using the formula a² – b² = (a + b) (a – b)
= 3x²(x + 2) (x – 2)
Third Polynomial = (2x⁵ – 2x⁴ – 4x³)
= 2x³(x² – x -2), by taking 2x³ common
= 2x³(x² – 2x + x – 2), by splitting the middle term -x = -2x + x
= 2x³(x(x – 2) + 1(x – 2))
= 2x³(x + 1) ( x – 2)
The common terms of (2x² – 4x), (3x⁴ – 12x²), and (2x⁵ – 2x⁴ – 4x³) = x(x – 2)
Extra common terms are 2, 3x(x + 2), 2x²(x + 1)
Therefore, the required L.C.M. = x(x – 2) * 2 * 3x(x + 2) * 2x²(x + 1)
= 12x⁴ (x – 2) (x + 2) (x + 1).
Example 2.
Find the L.C.M of 3y³ – 18y²x + 27yx², 4y⁴ + 24y³x + 36y²x² and 6y⁴- 54y²x² by factorization.
Solution:
Given polynomials are 3y³ – 18y²x + 27yx², 4y⁴ + 24y³x + 36y²x² and 6y⁴- 54y²x²
First polynomial = 3y³ – 18y²x + 27yx²
= 3y(y² – 6yx +9x²) by taking 3y common
= 3y(y² – 3yx -3yx +9x²) by splitting the middle term -6yx = -3yx – 3yx
= 3y(y(y – 3x) -3x(y – 3x))
= 3y(y – 3x) (y – 3x)
Second Polynomial = 4y⁴ + 24y³x + 36y²x²
= 4y²(y² + 6yx + 9x²), by taking 4y² common
= 4y²(y² + 3yx + 3yx + 9x²) by splitting the middle term 6yx = 3yx + 3yx
= 4y²(y(y + 3x) + 3y(y + 3x))
= 4y²(y + 3x) (y + 3x)
Third Polynomial = 6y⁴- 54y²x²
= 6y²(y² – 9x²), by taking 6y² common
= 6y²(y² – (3x)²) by using a² – b² formula
= 6y²(y + 3x) (y – 3x)
The common factors of the above three expressions is ‘y’ and other common factors of first and third expressions are ‘3’ and ‘(y – 3x)’. The common factors of second and third expressions are ‘2’, ‘y’ and ‘(y + 3x)’. Other than these, the extra common factors in the first expression is ‘(y – 3x)’ and in the second expression are ‘2’ and ‘(y + 3x)’
Therefore, the required L.C.M. = y × 3 × (y – 3x) × 2 × y × (y + 3x) × (y – 3x) × 2 × (y + 3x)
= 12y²(y + 3x)² (y – 3x)²
Example 3.
Claculate the L.C.M of 2x² -x -1 and 4x² + 8x +3?
Solution:
Given polynomials are 2x² -x -1 and 4x² + 8x +3.
First Polynomial = 2x² -x -1
= 2x² – 2x + x -1, by splitting the middle term -x = -2x + x
= 2x(x – 1) + 1 (x – 1)
= (x – 1) (2x + 1)
Second Polynomial = 4x² + 8x +3
= 4x² + 6x + 2x + 3, by splitting the middle term 8x = 6x + 2x
= 2x(2x + 3) + 1(2x + 3)
= (2x + 1)(2x + 3)
The common factors of the above two expressions is (2x + 1). The extra common factor are (2x + 3), (x – 1).
Therefore, the required L.C.M. = (2x + 1) * (x – 1) * (2x + 3)
= (2x + 1) (x – 1) (2x + 3)