How To Find Lcm Of Algebraic Fractions

LCM of Algebraic Expressions

To calculate the LCM (Lowest Common Multiple) of the given algebraic expressions, we should convert the algebraic expressions into their simplest factors. And, so the product of their common factors and the remaining factors will give the LCM of the given algebraic expressions.

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∴ LCM = Common factors × Remaining factors

For example: 10^two + 5x + half-dozen and x² + 3x + 2 are two algenraic expressions. x² + 5x + 6 = (x + 2)(x + 3) and x² + 3x + two = (x + 1)(x+2) are their simplest factors. And, (x + ane)(x + 2)(x + iii) is the production of their common and remaining factors. And so, LCM of 10^two + 5x + six and 10^two + 3x + 2 is (x + 1)(x + 2)(x + 3).

While finding LCM, we should use the following steps:

Steps :

i. Factorize the given algebraic expressions.

two. Take out the common factors and and so the remaining factors.

3. The production of mutual and remaining factors will requite the required LCM.

The concept of LCM will be more clear from the following worked-out examples.

Worked Out Examples

Example ane: Find the LCM of 3x²yz, 4y²z and 5xz²

Solution:

Here,

one^st expression = 3x²yz = 3 × 10 × x × y × z

2^nd expression = 4y²z = 2 × 2 × y × y × z

three^rd expression = 5xz² = 5 × x × z × z

∴ LCM = three × 2 × 2 × 5 × x × x × y × y × z × z

= 60x²y^twoz²

Example 2: Find the LCM of m⁴ – 4mⁱⁱ and 3m² + 6m

Solution:

Here,

one^st expression = m^iv – 4mⁱⁱ

= m²(m2 – 4)

= grand²(m² – 2²)

= m × m(m + two)(grand – two)

2^nd expression = 3m² + 6m

= 3m(m + ii)

∴ LCM = 3m × m(m + 2)(yard – two)

= 3m²(g + 2)(m – 2)

Example 3: Detect the LCM of a² – 3a + two, a⁴ + a³ – 6a² and a³ + 2a² – 3a

Solution:

Here,

1^st expression = aⁱⁱ – 3a + 2

= a² – (2 + i)a + 2

= a² – 2a – a + ii

= a(a – 2) – 1(a – 2)

= (a – 2)(a – 1)

2^nd expression = a^iv + a³ – 6a²

= a²(a² + a – half dozen)

= a²{a² + (3 – 2)a – 6}

= aⁱⁱ(a² + 3a – 2a – 6)

= aⁱⁱ{a(a + three) – 2(a + iii)}

= a × a(a + three)(a – ii)

3^rd expression = a³ + 2a² – 3a

= a(a² + 2a – iii)

= a{a² + (3 – 1)a – 3}

= a(a^two + 3a – a – 3)

= a{a(a + iii) – ane(a + 3)}

= a(a + 3)(a – 1)

∴ LCM = a × a(a – one)(a – 2)(a + 3)

= aⁱⁱ(a – 1)(a – 2)(a + 3)

Example 4: Observe the LCM of ten² – 5x + 6, x² + 4x – 12 and x² – 2x

Solution:

Here,

ane^st expression = x² – 5x + 6

= x² – (3 + 2)x + 6

= x² – 3x – 2x + 6

= x(ten – 3) – 2(x – 3)

= (x – 3)(ten – ii)

2^nd expression = x² + 4x – 12

= xⁱⁱ + (vi – 2)x – 12

= x² + 6x – 2x - 12

= 10(ten + 6) – ii(x + 6)

= (x + 6)(x – 2)

three^rd expression = tenⁱⁱ – 2x

= 10(x – 2)

∴ LCM = x(x – two)(ten – 3)(x + half-dozen)

Example 5: Find the LCM of a² – b² – 2bc – c^two, b² – cⁱⁱ – 2ca – a² and c² – a² – 2ab – b²

Solution:

Here,

ane^st expression = a² – b² – 2bc – c²

= a^two – (b² + 2bc + c^two)

= aⁱⁱ – (b + c)^two

= (a + b + c)(a – b – c)

two^nd expression = b^two – cⁱⁱ – 2ca – a²

= b² – (c² + 2ca + a²)

= b^two – (c + a)²

= (b + c + a)(b – c – a)

= (a + b + c)(b – c – a)

three^rd expression = cⁱⁱ – a^two – 2ab – b^two

= c² – (a² + 2ab + bⁱⁱ)

= c² – (a + b)²

= (c + a + b)(c – a – b)

= (a + b + c)(c – a – b)

∴ LCM = (a + b + c)(a – b – c)(b – c – a)(c – a – b)

Example 6: Find the LCM of a³ – 2a^twob + 2ab² – bⁱⁱⁱ, a^four + b⁴ + a²b^two and 4a^ivb + 4ab⁴

Solution:

Here,

1^st expression = a³ – 2aⁱⁱb + 2ab^two – b³

= a³ – b³ – 2a²b + 2ab²

= (a – b)(a² + ab + bⁱⁱ) – 2ab(a – b)

= (a – b)(a^two + ab + b² – 2ab)

= (a – b)(a² – ab + b²)

2^nd expression = a^four + b⁴ + a²b²

= (aⁱⁱ)² + 2a²b² + (b^two)² – a²bⁱⁱ

= (a^two + b^two)² –(ab)^two

= (a² + b² + ab)(a² + b² – ab)

= (a² + ab + b²)(aⁱⁱ – ab + b²)

iii^rd expression = 4a⁴b + 4ab⁴

= 4ab(a³ + b³)

= 4ab(a + b)(a² – ab + b²)

∴ LCM = 4ab(a + b)(a – b)(a^two + ab + b^two)(a² – ab + b^two)

Example 7: Find the LCM of a² – 18a – xix + 20b – b², a² + a – bⁱⁱ + b and 4a^two – 4b^two + 8b - iv

Solution:

Here,

ane^st expression = a² – 2.a.9 + nine² – 100 + 20b – b²

= (a – ix)ⁱⁱ – (10^two – ii.10.b + b²)

= (a – 9)² – (ten – b)²

= {(a – 9) + (10 – b)}{(a – 9) – (x – b)}

= (a – nine + ten – b)(a – 9 – 10 + b)

= (a – b + ane)(a + b – 19)

two^nd expression = a² + a – b^two + b

= aⁱⁱ – b^two + a + b

= (a + b)(a – b) + 1(a + b)

= (a + b)(a – b + i)

iii^rd expression = 4a² – 4b² + 8b - 4

= 4(a² – b² + 2b – 1)

= iv{a² – (b^two – 2b + 1)}

= 4{a² – (b – 1)²}

= 4(a + b – ane)(a – b + 1)

∴ LCM = 4(a + b)(a + b – ane)(a – b + 1)(a + b – xix)

Example eight: Find the LCM of 1 + 4a + 4a² – 16a⁴, 1 + 2a – 8a³ – 16a⁴ and 1 – 8aⁱⁱⁱ

Solution:

Here,

1^st expression = 1 + 4a + 4a² – 16a^four

= 1^two + 2.1.2a + (2a)² – (4a²)^two

= (1 + 2a)² – (4aⁱⁱ)^two

= (1 + 2a + 4a^two)(one + 2a – 4a²)

two^nd expression = 1 + 2a – 8a³ – 16a⁴

= 1(i + 2a) – 8a³(ane + 2a)

= (1 + 2a)(1 – 8a³)

= (one + 2a){1³ – (2a)³}

= (1 + 2a)(1 – 2a)(1 + 2a + 4a²)

3^rd expression = 1 – 8a³

= 1³ – (2a)³

= (1 – 2a)(ane + 2a + 4a²)

∴ LCM = (1 + 2a)(1 – 2a)(1 + 2a + 4a²)(1 + 2a – 4aⁱⁱ)

If you have whatever question or problems regarding the LCM of algebraic expressions, you can ask hither, in the comment section below.

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How To Find Lcm Of Algebraic Fractions,

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