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 18a2c2.
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 + 12a2b2c2.
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.