Dividing Polynomials

Quotient of a Monomial by a Monomial

To divide a monomial by a monomial divide numerical coefficient by numerical coefficient, divide powers of same variable using second law of exponents. See second law of exponents

Examples:

(35x³ )/(7x) = 5x²

(16x² y² )/(8xy² ) = 2x

(Remark: Any nonzero number divided by itself is one. Hence y² /y² = 1. If you use law 2 you obtain y to the zero power. Any nonzero number to the zero power is defined to be one.

42x/(7x³ ) = 6/x²

Remark: If you reduce the fraction x/x³ obtain 1/x² . If Law 2 is used you obtain x to the negative two power. Since any nonzero number to a negative exponent is defined to be one divided by the nonzero number to the positive exponent obtained by changing the sign of the exponent, the two results agree. See discussion of negative exponents in any algebra text.

Quotient of a Polynomial by a Monomial

To divide a polynomial by a monomial divide each of the terms of the polynomial by a monomial.

Example:

(16x³ - 12x² + 4x)/(2x) = (16x³ )/(2x) - (12x² )/(2x) + (4x)/(2x) = 8x² - 6x + 2

Quotient of a Polynomial by a Polynomial

To divide a polynomial by a polynomail we use a long division technique similar to the long division technique used in arithmetic. The long division technique can not currently be illustrated adequately using HTML (Hyper Text Markup Language). Consult any algebra text for a discussion of this technique.

Remember in starting the long division process

1. Write dividend and divisor in terms of descending powers of variable leaving space for any missing powers of the variable or writing in the missing powers with coefficient zero. (If there is more than one variable, arrange dividend and divisor in terms of descending powers of one of the variables. This is beyond the scope of Math 390 at MC.)
2. Divide first term of divisor into first term of dividend (On subsequent iterations into first term of difference). Place this answer above long division symbol.
3. Multiply divisor by the expression just written above division symbol and align like terms.
4. Subtract line just written from line immediately above it. Remember to subtract we change the sign of the subtrahend and add or add the opposite.
5. Repeat steps 2 through 4 until the difference you obtain is a polynomial of degree less than the degree of the divisor.
6. If difference is zero division is exact. The quotient is the polynomial given accross the top. If difference in nonzero division is not exact, the quotient is the polynomial given accross the top plus the remainder (polynomial in last line) divided by the divisor.

Example:

(5x + 4x³ - 4)/(2x - 1)

                     2x²  +  x  + 3  + (-1)/(2x - 1)
                    ------------------------------------
      2x - 1   ¦  4x³  + 0x²  + 5x - 4     
                    4x³  -  2x²
                    -----------------------
                              2x²  + 5x - 4
                              2x²  -   x
                              ---------------
                                        6x - 4 
                                        6x - 3
                                        ------
                                             - 1 

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