Question

Find the LCM of the following

Answer 1

Least Common Multiple(LCM): The least common multiple of two or more algebraic expressions is the expression of lowest degree which is divisible by each of them without remainder.

LCM OF POLYNOMIALS :
1•Find the LCM of the numerical coefficient of the polynomials.
2•Factorise the given polynomials.
3•Take the highest power of each of the factors (including the ones in common)]
4•The product of the number and the powers of the factors obtained in step 1 and 3 is the LCM of the given polynomials.

SOLUTION :

•10 (9x² + 6 x y + y²) = 2 x 5 (9 x² + 6 x y + y²)
= 2 x 5 ((3 x)² + 2×3x y + (y)²)
= 2 x 5¹ x (3 x + y)²
[a² +2ab + b² = (a+b)²]

•12 (3x² - 5 xy - 2y²) = 2² x 3 (3 x² - 6 x y + x y - 2y²)
[By middle term splitting]
= 2² x 3 x [3 x (x - 2y) + y (x - 2y)]
= 2² x 3¹ x (3 x + y) (x - 2y)


•14 (6 x⁴ + 2 x³) = 2 x 7 x 2 x³ (3 x + 1)
= 2² x 7¹ x x³ (3 x + 1)

L.C.M = 2² x 5¹ x 7¹x 3¹ x x³ x (3 x + y)²(3 x + 1)(x - 2y)
[On taking the highest power of each of the factors (including the ones in common)]
L.C.M= 420 x³ (3 x + y)²(3 x + 1)(x - 2y)

Hence, the L.C.M is 420 x³ (3 x + y)²(3 x + 1)(x - 2y)

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Answer 2

Solution:-

given by:-
$p(x) = 10(9 {x}^{2} + 6xy + {y}^{2} ) \\ = 2 \times 5( {(3x)}^{2} + 2 \times 3x \times y + {y}^{2} ) \\ = 2 \times 5{(3x + y)}^{2} \\ q(x) = 12(3 {x}^{2} - 5xy - 2 {y}^{2} ) \\ 12(3 {x}^{2} - 6xy + xy - 2 {y}^{2} ) \\ 12(3x(x - 2y) + y(x - y)) \\ = 2 \times 2 \times 3 \times (x - 2y)(3x + y) \\ r(x) = 14(6 {x}^{4} + 2 {x}^{3} ) \\ = 2 \times 2 \times 7 \times {x}^{3} \times (3x + 1) \\ \\ lcm = 4 \times 3 \times 5 \times 7 \times {x}^{3} \times {(3x + y)}^{2} (x - 2y)((3x + 1) \\ lcm = 420 {x}^{3} {(3x + y)}^{2} (x - 2y)(3x + 1) \: ans$
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