H.C.F. of Polynomials by Factorization

Learn how to solve H.C.F. of polynomials by factorization splitting the middle term.

Solved examples on highest common factor of polynomials by factorization:

1. Find out the H.C.F. of x² - 3x - 18 and x² + 5x + 6 by factorization.

Solution:

First expression = x² - 3x - 18

                      = x² - 6x + 3x - 18, by splitting the middle term - 3x = - 6x + 3x

                      = x(x - 6) + 3(x - 6)

                      = (x - 6) (x + 3)

Second expression = x² + 5x + 6

                          = x² + 3x + 2x + 6, by splitting the middle term 5x = 3x + 2x

                          = x(x + 3) + 2(x + 3)

                          = (x + 3) (x + 2)

Therefore, in the two polynomials (x + 3) is the only common factors, so, the required H.C.F. = (x + 3).

2. Find out the H.C.F. of (2a² - 8b²), (4a² + 4ab - 24b²) and (2a² - 12ab + 16b²) by factorization.

Solution:

First expression = (2a² - 8b²)

                      = 2(a² - 4b²), by taking common 2

                      = 2[(a)² - (2b)²], using the identity of a² – b²

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

                      = 2 × (a + 2b) × (a - 2b)

Second expression = (4a² + 4ab - 24b²)

                          = 4(a² + ab - 6b²), by taking common 4

                      = 4(a² + 3ab - 2ab - 6b²), by splitting the middle term ab = 3ab - 2ab

                         = 4[a(a + 3b) - 2b(a + 3b)]

                         = 4(a + 3b) (a - 2b)

                         = 2 × 2 × (a + 3b) × (a - 2b)

Third expression = (2a² - 12ab + 16b²)

                      = 2(a² - 6ab + 8b²), , by taking common 2

              = 2(a² - 4ab - 2ab + 8b²), by splitting the middle term - 6ab = - 4ab - 2ab

                      = 2[a(a - 4b) - 2b(a - 4b)]

                      = 2(a - 4b) (a - 2b)

                      = 2 × (a - 4b) × (a - 2b)

From the above three expressions ‘2’ and ‘(a - 2b)’ are the common factors of the expressions.

Therefore, the required H.C.F. is 2 × (a - 2b) = 2(a - 2b)

8th Grade Math Practice

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