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 m^{3} – 3m^{2} + 2m and m^{3} + m^{2} – 6m by factorization.= m[m(m  2)  1(m  2)]
= m(m  2) (m  1)
= m × (m  2) × (m  1)
= 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)
= 3a[a(a  3x)  3x(a  3x)]
= 3a(a  3x) (a  3x)
= 3 × a × (a  3x) × (a  3x)
= 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) = 12a^{2}(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(a^{2}  4), 6(a^{2}  a  2) and 12(a^{2} + 3a  10) by factorization.= 6[a(a  2) + 1(a  2)]
= 6(a  2) (a + 1)
= 2 × 3 × (a  2) × (a + 1)
Third expression = 12(a^{2} + 3a  10)= 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)
8th Grade Math Practice
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