Taking out Common Factor from Polynomial
Algorithm for Taking out the Greatest Common Factor from Polynomial
- Find the greatest common factor (GCF) for the terms of the polynomial.
For numerical coefficients, the common factor is the greatest common divisor; for exponents with the same base, the common factor has the smallest exponent. - Take the common factor out. Divide all terms of the polynomial by that factor, and put the result in parentheses (how to divide by a monomial, see §18 of this Guide).
Notice! The correctness of the factorization can be checked by multiplying the resulting factors. Learn how to multiply monomials in your mind, and you won't make mistakes.
Examples of Problem Solving
Example 1. Take out the common factor:
a) $ abc-2a^3+ac^2 = a(bc+2a^2+c^2 )$
b) $ 16x^2 y+24xyz-8xy^2 = 8xy(2x+3z-y)$
c) $17ax-51a^2 x+34x = 17x(a-3a^2+2)$
d) $ab-a^2+ac = a(b-a+c)$
Example 2. Factorize the expression:
a) 2x(x+1)-3(x+1) = (2x-3)(x+1)
b) $ a(x^2+y^2 )+b(x^2+y^2 ) = (a+b)(x^2+y^2 )$
c) a(b-c)-d(c-b) = a(b-c)+d(b-c) = (a+d)(b-c)
d) x(y-z)-y+z = x(y-z)-(y-z) = (x-1)(y-z)
e) $2(x+y)(x-y)-(x+y)^2 = (x+y)(2(x-y)+(x+y) ) = (x+y)(2x-2y+x+y) = (x+y)(3x-y)$
f) $5a(a-b)^2-7a^2 (b-a) = 5a(a-b)^2+7a^2 (a-b) = a(a-b)(5(a-b)+7a) = a(a-b)(12a-5b) $
Example 3. Evaluate the expression:
a) 168,5x(x-y)+168,5y(x-y), if x = 1,y = -1
168,5x(x-y)+168,5y(x-y) = 168,5(x+y)(x-y)
Substitute: 168,5(1-1)(1+1) = 0
b) 6,48xy-$x^2$, if x = 5,48 and y = 1
6,48xy-$x^2$ = x(6,48y-x)
Substitute: 5,48 ∙(6,48∙1-5,48 ) = 5,48
Example 4. Take out the numerical coefficient:
a) $(4x+8)^3 = (4(x+2) )^3 = 4^3 (x+2)^3 = 64(x+2)^3$
b) $(15-5a)^4 = (5(3-a) )^4 = 5^4 (3-a)^4 = 625(3-a)^4$
Example 5. Take out the greatest common factor:
a) $(xy^2-y^3 )^3 = (y^2 (x-y) )^3 = y^6 (x-y)^3$
b) $(0,4a^3-0,7a^5 )^2 = (0,1a^3 (4-7a^2 ) )^2 = 0,01a^6 (4-7a^2 )^2$
