Relation Between H.C.F. and L.C.M. of Two Polynomials

The relation between H.C.F. and L.C.M. of two polynomials is the product of the two polynomials is equal to the product of their H.C.F. and L.C.M.

If p(x) and q(x) are two polynomials, then p(x) ∙ q(x) = {H.C.F. of p(x) and q(x)} x {L.C.M. of p(x) and q(x)}.

1. Find the H.C.F. and L.C.M. of the expressions a² – 12a + 35 and a² – 8a + 7 by factorization.

Solution:

First expression = a² – 12a + 35

                      = a² – 7a – 5a + 35

                      = a(a – 7) – 5(a – 7)

                      = (a – 7) (a – 5)


Second expression = a² – 8a + 7

                          = a² – 7a – a + 7

                          = a(a – 7) – 1(a – 7)

                          = (a – 7) (a – 1)

Therefore, the H.C.F. = (a – 7) and L.C.M. = (a – 7) (a – 5) (a – 1)

Note:

(i) The product of the two expressions is equal to the product of their factors.

(ii) The product of the two expressions is equal to the product of their H.C.F. and L.C.M.

Product of the two expressions = (a² – 12a + 35) (a² – 8a + 7)

                                           = (a – 7) (a – 5) (a – 7) (a – 1)

                                           = (a – 7) (a – 7) (a – 5) (a – 1)

                                           = H.C.F. × L.C.M. of the two expressions

2. Find the L.C.M. of the two expressions a² + 7a – 18, a² + 10a + 9 with the help of their H.C.F.

Solution:

First expression = a² + 7a – 18

                      = a² + 9a – 2a – 18

                      = a(a + 9) – 2(a + 9)

                      = (a + 9) (a – 2)

Second expression = a² + 10a + 9

                          = a² + 9a + a + 9

                          = a(a + 9) + 1(a + 9)

                          = (a + 9) (a + 1)

Therefore, the H.C.F. = (a + 9)

Therefore, L.C.M. = Product of the two expressions/H.C.F.

                        = \(\frac{(a^{2} + 7a - 18) (a^{2} + 10a + 9)}{(a + 9)}\)

                        = \(\frac{(a + 9) (a - 2) (a + 9) (a + 1)}{(a + 9)}\)

                        = (a – 2) (a + 9) (a + 1)

3. m² – 5m -14 is an expression. Find out another similar expression such that their H.C.F. is (m – 7) and L.C.M. is m³ – 10m² + 11m + 70.

Solution:

According to the problem,

Required Expression = \(\frac{L.C.M.  ×  H.C.F.}{Given expression}\)

                           = \(\frac{(m^{3}  -  10m^{2}  +  11x  +  70)(x  -  7)}{x^{2}  -  5x  -  14}\)

                           = \(\frac{(m^{2} - 5m - 14)(x  -  5)(x  -  7)}{x^{2}  -  5x  -  14}\)

                            = (m – 5)(m – 7)

                            = m² – 12m + 35





Therefore, the required expression = m² – 12m + 35

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8th Grade Math Practice

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