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```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 18a2c2.

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 + 12a2b2c2.

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.
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