# L.C.M. of Polynomials by Factorization

Learn how to solve L.C.M. of polynomials by factorization splitting the middle term.

Solved examples on lowest common multiple of polynomials by factorization:

1. Find the L.C.M of m3 – 3m2 + 2m and m3 + m2 – 6m by factorization.

Solution:

First expression = m3 – 3m2 + 2m

= m(m2 – 3m + 2), by taking common ‘m’

= m(m2 - 2m - m + 2), by splitting the middle term -3m = -2m - m

= m[m(m - 2) - 1(m - 2)]

= m(m - 2) (m - 1)

= m × (m - 2) × (m - 1)

Second expression = m3 + m2 – 6m

= m(m2 + m - 6) by taking common ‘m’

= m(m2 + 3m – 2m - 6), by splitting the middle term m = 3m - 2m

= m[m(m + 3) - 2(m + 3)]

= m(m + 3)(m - 2)

= m × (m + 3) × (m - 2)

In both the expressions, the common factors are ‘m’ and ‘(m - 2)’; the extra common factors are (m - 1) in the first expression and (m + 3) in the 2nd expression.

Therefore, the required L.C.M. = m × (m - 2) × (m - 1) × (m + 3)

= m(m - 1) (m - 2) (m + 3)

2. Find the L.C.M of 3a3 - 18a2x + 27ax2, 4a4 + 24a3x + 36a2x2 and 6a4 - 54a2x2 by factorization.

Solution:

First expression = 3a3 -18a2x + 27ax2

= 3a(a2 - 6ax + 9x2), by taking common ‘3a’

= 3a(a2 - 3ax - 3ax + 9x2), by splitting the middle term - 6ax = - 3ax - 3ax

= 3a[a(a - 3x) - 3x(a - 3x)]

= 3a(a - 3x) (a - 3x)

= 3 × a × (a - 3x) × (a - 3x)

Second expression = 4a4 + 24a3x + 36a2x2

= 4a2(a2 + 6ax + 9x2), by taking common ‘4a2

= 4a2(a2 + 3ax + 3ax + 9x2), by splitting the middle term 6ax = 3ax + 3ax

= 4a2[a(a + 3x) + 3x(a + 3x)]

= 4a2(a + 3x) (a + 3x)

= 2 × 2 × a × a × (a + 3x) × (a + 3x)

Third expression = 6a4 - 54a2x2

= 6a2(a2 - 9x2), by taking common ‘6a2

= 6a2[(a)2 - (3x)2), by using the formula of a2 – b2

= 6a2(a + 3x) (a - 3x), we know a2 – b2 = (a + b) (a – b)

= 2 × 3 × a × a × (a + 3x) × (a - 3x)

The common factors of the above three expressions is ‘a’ and other common factors of first and third expressions are ‘3’ and ‘(a - 3x)’.

The common factors of second and third expressions are ‘2’, ‘a’ and ‘(a + 3x)’.

Other than these, the extra common factors in the first expression is ‘(a - 3x)’ and in the second expression are ‘2’ and ‘(a + 3x)’

Therefore, the required L.C.M. = a × 3 × (a - 3x) × 2 × a × (a + 3x) × (a - 3x) × 2 × (a + 3x) = 12a2(a + 3x)2(a - 3x)2

More problems on L.C.M. of polynomials by factorization splitting the middle term:

3. Find the L.C.M. of 4(a2 - 4), 6(a2 - a - 2) and 12(a2 + 3a - 10) by factorization.

Solution:

First expression = 4(a2 - 4)

= 4(a2 - 22), by using the formula of a2 – b2

= 4(a + 2) (a - 2), we know a2 – b2 = (a + b) (a – b)

= 2 × 2 × (a + 2) × (a - 2)

Second expression = 6(a2 - a - 2)

= 6(a2 – 2a + a - 2), by splitting the middle term – a = – 2a + a

= 6[a(a - 2) + 1(a - 2)]

= 6(a - 2) (a + 1)

= 2 × 3 × (a - 2) × (a + 1)

Third expression = 12(a2 + 3a - 10)

= 12(a2 + 5a – 2a - 10), by splitting the middle term 3a = 5a – 2a

= 12[a(a + 5) - 2(a + 5)]

= 12(a + 5) (a - 2)

= 2 × 2 × 3 × (a + 5) × (a - 2)

In the above three expressions the common factors are 2 and (a - 2).

Only in the second expression and third expression the common factor is 3.

Other than these, the extra common factors are (a + 2) in the first expression, (a + 1) in the second expression and 2, (a + 5) in the third expression.

Therefore, the required L.C.M. = 2 × (a - 2) × 3 × (a + 2) × (a + 1) × 2 × (a + 5)

= 12(a + 1) (a + 2) (a - 2) (a + 5)

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