Division of Monomial or Polynomial by Monomial
Rule for Division of Polynomial by Monomial
To divide a polynomial by a monomial, divide each term of the polynomial by the given monomial and sum up the results.
The quotient of a polynomial and a monomial is not always a polynomial.
For example, the polynomial (2x+3y) is not divisible by the monomial xz so that the quotient is a polynomial (that is, an integer expression, see §1 of this Guide).
When dividing by a monomial, it is assumed that its variables take such values that the monomial is not equal to 0.
Examples of Problem Solving
Example 1. Find the quotient:
a) $ (3ax+5a-8a^2 ):(2a) = \frac{3ax+5a-8a^2}{2a} = \frac{3ax}{2a}+ \frac{5a}{2a}- \frac{8a^2}{2a} = \frac{3x}{2}+ \frac{5}{2}-4a = 1,5x+2,5-4a $
b) $ (5xy-x^2 z+3x):x = \frac{5xy-x^2 z+3x}{x} = \frac{5xy}{x}- \frac{x^2 z}{x} + \frac{3x}{x} = 5y-xz+3 $
c) $ (16x^3 y-12x^2 y^2 ):(4xy) = \frac{16x^3 y-12x^2 y^2}{4xy} = \frac{16x^3 y}{4xy}- \frac{12x^2 y^2}{4xy} = 4x^2-3xy $
d) $ (27ab^2-15a^3 b):(3ab)= \frac{27ab^2-15a^3 b}{3ab} = \frac{27ab^2}{3ab}- \frac{15a^3 b}{3ab} = 9b-5a^2 $
Example 2. Simplify the expression:
a) (6ax+5x):x-(5ax+6a):a = (6a+5)-(5x+6) = 6a-5x-1
b) $ (a^2 b-ab^2+3ab):(\frac{1}{2} ab)+(6b-18):3 = (2a-2b+6)+(2b-6) = 2a $
Example 3. Evaluate the expression for x=5
a) $ (\frac{x}{2}+x^2 ):x-(x^3+x):(2x) = (\frac{1}{2}+x)-(\frac{x^2}{2}+ \frac{1}{2}) = x- \frac{x^2}{2} $
Substitute: 5- $\frac{5^2}{2}$ = -7,5
b) $ (ax^2+14 ax^3 ):(ax^2 )-(13ax^2+5ax):(ax) = (1+14x)-(13x+5) = x-4 $
Substitute: 5-4 = 1
Example 4. Solve the equation:
a) $ (3x^4+ \frac{1}{3} x^2 ):x-7x^3:(3x^2 ) = 3x^3+8 $
$ 3x^3+ \frac{1}{3} x- \frac{7}{3} x = 3x^3+8 $
-2x = 8
x = -4
b) $ (\frac{1}{5} x^4-3x):(4x)-(8x^5+x^4 ):x^4 = -0,16 $
$ \frac{x}{20}- \frac{3}{4}-1 = -0,16 $
0,05x-8x = -0,16+1,75
-7,95x = 1,59
x = -0,2