Highest Common Factor of Polynomials by Factorization

How to find the highest common factor of polynomials by factorization?

Let us follow the following examples to know how to find the highest common factor (H.C.F.) or greatest common factor (G.C.F.) of polynomials by factorization.

Solved examples of highest common factor of polynomials by factorization:

1. Find out the H.C.F. of a²b + ab² and a²c + abc by factorization.

Solution:

First expression = a²b + ab²

                      = ab(a + b)

                      = a × b × (a + b)

Second expression = a²c + abc

                          = ac(a + b)

                          = a × c × (a + b)

It can be seen, in both the expressions ‘a’ and ‘(a + b)’ are the common factors and there is no other common factor.

Therefore, the required H.C.F. a²b + ab² and a²c + abc is a(a + b)



2. Find the out the H.C.F. of (a²b + a²c) and (ab + ac)² by factorization.

Solution:

First expression = a²b + a²c

                      = a²(b + c)

                      = a × a × (b + c)

Second expression = (ab + ac)²

                          = (ab + ac) (ab + ac)

                          = a(b + c) a(b + c)

                          = a × a × (b + c) × (b + c)    

It can be seen that, in both the expressions ‘a’, ‘a’ and ‘(b + c)’ are the common factors and there is no other common factor.

Therefore, the required H.C.F. is a × a × (b + c) = a²(b + c).



3. Find the out the H.C.F. of c(a + b)², (a²c² – b²c²) and a(ac² + bc²) by factorization.

Solution:

First expression = c(a + b)²

                      = c × (a + b) × (a + b)

Second expression = (a²c² - b²c²)

                          = c²(a² - b²)

                          = c²(a + b) (a - b)

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

Third expression = a(ac² + bc²)

                       = ac²(a + b)

                       = a × c × c × (a + b)

It can be seen that, c and (a + b) are the common factors of the expressions.

Therefore, the required H.C.F. of c(a + b)², (a²c² – b²c²) and a(ac² + bc²) is c(a + b)



4. Find out the H.C.F. of 3x²(y + z)² and 6x(y² - z²) by factorization.

Solution:

First expression = 3x²(y + z)²

                      = 3x² (y + z)(y + z)

                      = 3 × x × x × (y + z) × (y + z)

Second expression = 6x(y² - z²)

                          = 6x(y² - z²)

                          = 6x(y + z) (y - z)

                          = 2 × 3 × x × (y + z) × (y - z)

Therefore, the required H.C.F. is 3 × x × (y + z) = 3x(y + z)

8th Grade Math Practice

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