Download division of polynomials

DIVISION OF POLYNOMIALS
Definition: Let N and D be two polynomials. The quotient of dividing N by D is
represented by the rational fraction N/D. If a polynomial Q such that N=DxQ exists, we
say that N is divisible by D, and that Q is the quotient of the division.
We will study three cases of division of polynomials. The first is the division of two
monomials, the second is the division of o polynomial by a monomial, and the third the
division of two polynomials.
Division of Two Monomials: To divide two monomials we divide their numerical coefficients and their
numerical parts, keeping in mind that to divide powers to the same base we subtract the exponents.
Example:
15 x 4 y 3 z 5
 5 x 2 yz 3
2 2 2
3x y z
Division of a Polynomial by a Monomial: To divide a polynomial by a monomial we
divide each term of the polynomial by the monomial. Example:
8x3 y 2  6 x 4 y  12 x3 y3 z 3 8x3 y 2 6 x 4 y 12 x3 y 3 z 3
3
 2  2 
 2 xy  x 2  3xy 2 z 3
2
2
4x y
4x y 4x y
4x y
2
Division of a Polynomial by a Polynomial: To divide two polynomials we first, if necessary, complete and
order them in descending order of exponents. Then we divide the first term of the dividend by he first term
of the divisor. Next we multiply this quotient by the entire divisor and we subtract the product from the
dividend. This process is repeated after bringing down the next term from the dividend until the degree of
the remainder is less than the degree of the divisor. Example:
12 x 3  4  8 x
2  3x
8
40
4 x2  x 
Quotient (whole part)
3
9
Divisor
3 x  2 12 x 3  0 x 2  8 x  4 Dividend (complete and ordered)
12 x 3  8 x 2
 8 x2  8 x
16
x
3
40
x4
3
40
80
 x
3
9
44

Remainder
9
8 x 2 
Algebraic Fractions. Properties: If M and N are two algebraic expressions, the quotient, represented by
M/N, is referred to as an algebraic fraction. If, as we pointed out before, M and N are polynomials, the
fraction is called rational. Examples:
x 3
x2  2 x  1
b
,
,
3
2
x 7
x y
a  b2
If both terms, numerator and denominator, of an algebraic fraction are multiplied or divided by a non-zero
expression, the fraction we get is equivalent to the first one. When we divide the numerator and
denominator of a fraction by a common factor to both, the process is called simplification. To simplify a
fraction it is helpful to factor both terms conveniently. The equivalent fraction, reduced to lowest terms, is
then obtained by canceling out all common factors from the numerator and the denominator. Examples:
x 2  9  x  3 ( x  3) x  3


3x  9
3( x  3)
3
6a 5b3 2  3a 3b 2b 3b 2


8a 3bc 2  4a 3bc
4c
The concept of equivalent fractions is also used to reduce two or more fractions to a common denominator.
This could be the product of all the denominators, although it is more convenient to use the smallest or least
common denominator (lcd). As we do with numbers, the lcd is obtained from the prime factorization of the
denominators by forming a single product with all common factors with the highest exponent and all noncommon factors. Example: Reduce the following fractions to a common denominator,
a
,
3
b
,
a  4a  4
2
1
a 4
2
33
a 2  4a  4  (a  2) 2
a 2  4  (a  2)(a  2)
The lcd is 3(a  2) 2 (a  2), and the equivalent fractions are,
a(a  2) 2 (a  2)
3b(a  2)
3(a  2)
,
, and
2
2
3(a  2) (a  2) 3(a  2) (a  2)
3(a  2) 2 (a  2)
Notice that each numerator is the result of multiplying the quotient of the lcd and each denominator, by the
corresponding numerator.
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EXERCISES
Perform the following divisions:
b) (12 p 2 r 2 )  ( 12 p 2 r 2 )
1. a) 15 x 2 y 2 z    5 xz 
2.
3.
8a b c  5a bc  6a b c   5a bc 
 6 x  7 x  8   x  x  1
3 3
3
2
4 2 2
2
2
2
x 2  16 y 2
x 2  8xy  16 y 2
x
y
z
,
,
5. Reduce to a common denominator: a)
4
6
36
4. Simplify:
a)
4 x5 y
2 x2
b)
b)
3
,
x2
2x 1
,
x2  x
x 1
x 1
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