RD Sharma Solutions Class 8 Factorization Exercise 7.1

Find the greatest common factor of the following polynomials (Q.1 – Q.14)

Q.1)  2x² and 12x²

Soln.:

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.

Q.2)  6x³y and 18x²y³

Soln.:

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.

Q.3)   7x, 21x² and 14xy²                       

Soln.:

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.

Q.4)   42x²yz and 63x³y²z³

Soln.:

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.

Q.5)   12ax² , 6a²x³ and 2a³x⁵

Soln.:

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.

Q.6)   9x² , 15x²y³ , 6xy² and 21x²y²

Soln.:

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.

Q.7)  4a²b³ , -12a³b , 18a⁴b³

Soln.:

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.

Q.8)  6x²y² , 9xy³ , 3x³y²

Soln.:

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.

Q.9)  a²b³ , a³b²

Soln.:

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.

Q.10)  36a²b²c⁴ , 54a⁵c² , 90a⁴b²c²

Soln.:

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.

Q.11)  x³ , -yx²

Soln.:

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.

Q.12)  15a³ , -45a² , -150a

Soln.:

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.

Q.13)  2x³y² , 10x²y³ , 14xy

Soln.:

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.

Q.14)  14x³y⁵ , 10x⁵y³ , 2x²y²

Soln.:

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.

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

Q.15)  5a⁴ + 10a³ – 15a²

Soln.:

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.

Q.16)  2xyz + 3x²y + 4y²

Soln.:

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.

Q.17)  3a²b² + 4b²c² + 12a²b²c^2.

Soln.:

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.