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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 x4 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 33 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. *************************************************************************************** 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 ***************************************************************************************