Lesson 3 · from §39 of the 1910 text
Long Division of Polynomials
Division undoes multiplication, so look at a multiplication first: (4x^2 - 5x + 6)(2x - 3) = 8x^3 - 22x^2 + 27x - 18. Now divide 8x^3 - 22x^2 + 27x - 18 by 4x^2 - 5x + 6. The quotient must be 2x - 3, and here's how we'd find it without knowing:
\begin{array}{r|l} 8x^3 - 22x^2 + 27x - 18 & 4x^2 - 5x + 6 \\ \hline \underline{8x^3 - 10x^2 + 12x} \phantom{{}- 18} & 2x - 3 \\ -12x^2 + 15x - 18 & \\ \underline{-12x^2 + 15x - 18} & \\ 0 & \end{array}
The first term of the quotient, 2x, is the highest term of the dividend divided by the highest term of the divisor: 8x^3 \div 4x^2. Multiply the whole divisor by 2x and subtract. Then repeat with the remainder: -12x^2 \div 4x^2 = -3. Nothing is left over, so the division is exact.
Rule. Arrange dividend and divisor in descending (or ascending) powers of the same letter. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Multiply the whole divisor by it, write the product under the dividend, and subtract. Divide the first term of the remainder by the first term of the divisor for the next term of the quotient, and repeat until nothing is left or the remainder's degree is lower than the divisor's.
If there's a remainder, write the answer like a mixed number in arithmetic, where \tfrac{17}{5} = 3\tfrac{2}{5} means 3 + \tfrac25: \frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}.
Check: divisor × quotient + remainder = dividend. Or substitute numbers, but never ones that make the divisor zero.
Quick check
Divide x^2 + 9x + 20 by x + 4.
From the history of algebra
Why 3\tfrac{2}{5} means addition. Writing two letters side by side, like ab, means multiplication. But in the mixed number 3\tfrac{2}{5}, writing side by side means addition. That habit comes from the Arabs, who showed every addition by placing the numbers one after another with no sign at all. The later Greeks wrote the same way.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Divide. Give the quotient and the remainder (0 if it divides exactly).
Divide x^2+7x+12 by x+3.
Divide x^2-2x-15 by x-5.
Divide x^2+5x+6 by x+3.
Divide -7x+6+x^2 by x-1.
Divide 6x^2-13x+6 by 2x-3.
Divide 25x^4+30x^2-7 by 7+5x^2.
Divide 12a^2-21+19a by 4a-3.
Divide -8+x^3+4x-2x^2 by x-2.
Divide a^3+3a^2b+3ab^2+b^3 by a+b.
Divide x^3-15x^2+65x-63 by x-7.
Divide 5x^2+5x-25x^3-1 by 5x^2-1.
Divide 2x^3-14x^2+14x+12 by 2x-4.
Divide 3a^3+28a^2+89a-140 by 3a-5.
Divide 37x+6x^3-24-23x^2 by 2x-3.
Divide 53a+8-53a^2+12a^3 by 4a^2-7a-1.
Divide 15a^3-56a^2+99a-70 by 3a^2-7a+10.
Divide 23a^2+a^4-55a+11a^3-140 by a^2-5.
Divide 4a^3+1+a^4+4a+6a^2 by 1+a^2+2a.
Divide a^4-8a^3+24a^2-32a+16 by a^2-4a+4.
Divide 40x-31x^2+21+x^4+4x^3 by x^2-3-7x.
Divide 11x-42x^2+10x^4-27x^3-36 by 9+2x^2-5x.
Divide x^4-3a^2x^2+a^4 by x^2-ax+a^2.
Divide a^4+4b^4+3a^2b^2 by 2b^2+a^2-ab.
Divide 16x^4-60x^2y^2+25y^4 by 4x^2-10xy-5y^2.
Divide 9a^4+49b^4+26a^2b^2 by 7b^2+3a^2+4ab.
Divide 4a^8-44a^4b^4+100b^8 by 2a^4-10b^4+2a^2b^2.
Divide 25x^3-10x^2+40x-18 by 5x-6.
Divide x^3-y^3 by x-y.
Divide a^3-125b^3 by a-5b.
Divide a^6+343b^3 by a^2+7b.
Divide x^4-16 by x+2.
Divide y^5-5y^2-3000 by y-5.
Divide x^4+y^4 by x-y.
Divide x^4+y^4 by x+y.
Divide x^4-y^4 by x+y.
Divide x^4-y^4 by x-y.
Divide x^5+y^5 by x+y.
Divide x^5+y^5 by x-y.
Divide x^5-y^5 by x+y.
Divide x^5-y^5 by x-y.
Divide x^{2a}-5x^{a}+6 by x^{a}-3.
Divide x^{6a}-7x^{3a}+12 by x^{3a}-4.
Divide x^{2a}+2x^{a+1}+3x^{a}+3x by x^{a}+2x.