To divide 9x² - 6x - 3 by 3, simply set the problem up as a fraction: (9x² - 6x - 3)/3. The numerator can be factored by 3, which is a common element: 3(3x² - 2x - 1)/3, which can then be canceled by the 3 in the denominator, yielding a solution of 3x² - 2x - 1. An alternative approach is to set up three separate fractions for each term in the numerator and solve: 9x²/3 - 6x/3 - 3/3 which also reduces to 3x² - 2x - 1.
The same techniques apply when dividing a polynomial by a variable. For example, when dividing 16x² + 4x by 2x, set up the fraction and simplify: (16x² + 4x)/2x = 4x(4x + 1 )/2x. Then 2x cancels out, and this is further reduced to 2(4x + 1). Setting up the separate fractions lets you double check the answer: 16x²/2x + 4x/2x reduces to 8x + 2 or 2(4x + 1).
A polynomial division problem can sometimes be solved by factoring. For example, in the problem (x² - 9)/(x + 3), factoring can yield a quick result, as the dividend is a difference of squares. The difference of squares, or x² - a², factors to (x + a)(x - a). Therefore, x² - 9 = (x + 3)(x - 3), making the fraction easily reduced: ((x + 3)(x - 3))/x + 3, the term (x + 3) cancels out, leaving the solution (x - 3). If you can easily factor the dividend, this method can sometimes yield a quick solution to a division problem.
Long division of polynomials works along the same principles as it does with numbers. You set up the division and solve for the first element of the polynomial. The order of exponents of the variable is important in this method, however. The largest exponent must be in the first place, with the descending exponents following in order. If there is a gap, for example, if you are given the dividend (8x³ + 6x + 4), an empty place holder must be set up for x². The dividend would then be in the form: 8x³ + 0x² + 6x + 4.