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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 β 3m2 + 2m and m3 + m2 β 6m by factorization.= m[m(m - 2) - 1(m - 2)]
= m(m - 2) (m - 1)
= m Γ (m - 2) Γ (m - 1)
= m[m(m + 3) - 2(m + 3)]
= m(m + 3)(m - 2)
= m Γ (m + 3) Γ (m - 2)
In both the expressions, the common factors are βmβ and β(m - 2)β; 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 Γ (m - 2) Γ (m - 1) Γ (m + 3)
= m(m - 1) (m - 2) (m + 3)
= 3a[a(a - 3x) - 3x(a - 3x)]
= 3a(a - 3x) (a - 3x)
= 3 Γ a Γ (a - 3x) Γ (a - 3x)
= 2 Γ 3 Γ a Γ a Γ (a + 3x) Γ (a - 3x)
The common factors of the above three expressions is βaβ and other common factors of first and third expressions are β3β and β(a - 3x)β.
The common factors of second and third expressions are β2β, βaβ and β(a + 3x)β.
Other than these, the extra common factors in the first expression is β(a - 3x)β and in the second expression are β2β and β(a + 3x)β
Therefore, the required L.C.M. = a Γ 3 Γ (a - 3x) Γ 2 Γ a Γ (a + 3x) Γ (a - 3x) Γ 2 Γ (a + 3x) = 12a2(a + 3x)2(a - 3x)2More 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.= 6[a(a - 2) + 1(a - 2)]
= 6(a - 2) (a + 1)
= 2 Γ 3 Γ (a - 2) Γ (a + 1)
Third expression = 12(a2 + 3a - 10)= 12[a(a + 5) - 2(a + 5)]
= 12(a + 5) (a - 2)
= 2 Γ 2 Γ 3 Γ (a + 5) Γ (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 Γ (a - 2) Γ 3 Γ (a + 2) Γ (a + 1) Γ 2 Γ (a + 5)
= 12(a + 1) (a + 2) (a - 2) (a + 5)
8th Grade Math Practice
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