Chapter 7: Factorization Exercise – 7.1

Question: 1

Find the greatest common factor of the polynomials

 2x² and 12x²

Solution:

The numerical coefficients of the given monomials are 2 and 12.

So, the greatest common factor of 2 and 12 is 2.

The common literal appearing in the given monomials is x.

The smallest power of x in the two monomials is 2.

The monomial of the common literals with the smallest powers is x².

Hence, the greatest common factor is 2x².

Question: 2

Find the greatest common factor of the polynomials

 6x³y and 18x²y³

Solution:

The numerical coefficients of the given monomials are 6 and 18.

The greatest common factor of 6 and 18 is 6.

The common literals appearing in the two monomials are x and y.

The smallest power of x in the two monomials is 2.

The smallest power of y in the two monomials is 1.

The monomial of the common literals with the smallest powers is x²y.

Hence, the greatest common factor is 6x²y.

Question: 3

Find the greatest common factor of the polynomials

7x, 21x² and 14xy²

Solution:

The numerical coefficients of the given monomials are 7, 21 and 14.

The greatest common factor of 7, 21 and 14 is 7.

The common literal appearing in the three monomials is x.

The smallest power of x in the three monomials is 1.

The monomial of the common literals with the smallest powers is x.

Hence, the greatest common factor is 7x.

Question: 4

Find the greatest common factor of the polynomials

42x²yz and 63x³y²z³

Solution:

The numerical coefficients of the given monomials are 42 and 63.

The greatest common factor of 42 and 63 is 21.

The common literals appearing in the two monomials are x, y and z.

The smallest power of x in the two monomials is 2.

The smallest power of y in the two monomials is 1.

The smallest power of z in the two monomials is 1.

The monomial of the common literals with the smallest powers is x²yz.

Hence, the greatest common factor is 21x²yz.

Question: 5

Find the greatest common factor of the polynomials

12ax², 6a²x³ and 2a³x⁵

Solution:

The numerical coefficients of the given monomials are 12, 6 and 2.

The greatest common factor of 12, 6 and 2 is 2.

The common literals appearing in the three monomials are a and x.

The smallest power of a in the three monomials is 1.

The smallest power of x in the three monomials is 2.

The monomial of common literals with the smallest powers is ax².

Hence, the greatest common factor is 2ax².

Question: 6

Find the greatest common factor of the polynomials

9x², 15x²y³, 6xy² and 21x²y²

Solution:

The numerical coefficients of the given monomials are 9, 15, 6 and 21.

The greatest common factor of 9, 15, 6 and 21 is 3.

The common literal appearing in the three monomials is x.

The smallest power of x in the four monomials is 1.

The monomial of common literals with the smallest powers is x.

Hence, the greatest common factor is 3x.

Question: 7

Find the greatest common factor of the polynomials

4a²b³, -12a³b, 18a⁴b³

Solution:

The numerical coefficients of the given monomials are 4, -12 and 18.

The greatest common factor of 4. -12 and 18 is 2.

The common literals appearing in the three monomials are a and b.

The smallest power of a in the three monomials is 2.

The smallest power of b in the three monomials is 1.

The monomial of the common literals with the smallest powers is a²b.

Hence. the greatest common factor is 2a²b.

Question: 8

Find the greatest common factor of the polynomials

6x²y², 9xy³, 3x³y²

Solution:

The numerical coefficients of the given monomials are 6, 9 and 3.

The greatest common factor of 6, 9 and 3 is 3.

The common literals appearing in the three monomials are x and y.

The smallest power of x in the three monomials is 1.

The smallest power of y in the three monomials is 2.

The monomial of common literals with the smallest powers is xy².

Hence, the greatest common factor is 3xy².

Question: 9

Find the greatest common factor of the polynomials

a²b³, a³b²

Solution:

The numerical literals in the three monomials are a and b.

The smallest power of x in the three monomials is 2.

The smallest power of y in the three monomials is 2.

The monomial of common literals with the smallest powers is a²b².

Hence, the greatest common factor is a²b².

Question: 10

Find the greatest common factor of the polynomials

36a²b²c⁴, 54a⁵c², 90a⁴b²c²

Solution:

The numerical coeff.  of the given monomials are 36, 54, and  90.

The greatest common factors of 36, 54, and 90 is 18.

The common literals appearing in the three monomials are a and c.

The smallest power of a in the three monomials is 2.

The smallest power of c in the three monomials is 2.

The monomial of common literals with the smallest powers is a²c².

Hence, the greatest common factor is 18a²c^2. 

Question: 11

Find the greatest common factor of the polynomials

 x³, – yx²

Solution:

The common literal appearing in the two monomials is X.

The smallest power of X in both the monomials is 2.

Hence, the greatest common factor is x².

Question: 12

Find the greatest common factor of the polynomials 

15a³, - 45a², -150a

Solution:

The numerical coeff. of the given monomials are -15, -45 and -150.

The greatest common factor of 15, -45 and -150 is 15.

The common literal appearing in the three monomials is a.

The smallest power of a in the three monomials is 1.

Hence, the greatest common factor is 15a.

Question: 13

Find the greatest common factor of the polynomials 

2x³y², 10x²y³, 14xy

Solution:

The numerical coeff. of the given monomials are 2, 10 and 14.

The greatest common factor of 2, 10 and 14 is 2.

The common literals appearing in the three monomials are x and y.

The smallest power of X in the three monomials is 1.

The smallest power of y in the three monomials is 1.

The monomials of common literals with the smallest power is xy.

Hence, the greatest common factor is 2xy.

Question: 14

Find the greatest common factor of the polynomials 

14x³y⁵, 10x⁵y³, 2x²y²

Solution:

The numerical coeff. of the given monomials are 14, 10  and 2.

The greatest common factor of 14, 10 and 2 is 2.

The common literals appearing in the three monomials are x and y.

The smallest power of X in the three monomials is 2.

The smallest power of Y in the three monomials is 2.

The monomials of common literals with the smallest powers is x²y².

Hence, the greatest common factor is 2x²y².

Question: 15

Find the greatest common factor of the terms in the following expressions:

 5a⁴ + 10a³ – 15a²

Solution:

The numerical coeff. of the given monomials are 5a⁴, 10a³, and 15a².

The greatest common factor of 5a⁴, 10a³, and 15a² is 5.

The common literal appearing in the three monomials is a.

The smallest power of a in the three monomials is 2.

The monomials of common literals with the smallest powers is a².

Hence, the greatest common factor is 5a².

Question: 16

Find the greatest common factor of the terms in the following expressions:

2xyz + 3x²y + 4y²

Solution:

The numerical coeff. of the given monomials are 2xyz, 3x²y and 4y².

The greatest factor of 2xyz, 3x²y and 4y² is 1.

The common literal appearing in the three monomials is y.

The smallest power of y in the three monomials is 1.

The monomials of common literals with the smallest power is y.

Hence, the greatest common factor is y.

Question: 17

Find the greatest common factor of the terms in the following expressions:

3a²b² + 4b²c² + 12a²b²c^2.

Solution:

The numerical coeff. of the given monomials are 3a²b², 4b²c² and 12a²b²c².

The greatest common factor of 3a²b², 4b²c² and 12a²b²c² is 1.

The common literal appearing in the three monomials is b.

The smallest power of b in the three monomials is 2.

The monomials of common literals with the smallest powers is b².

Hence, the greatest common factor is b².