Lesson 2 · from §38 of the 1910 text
Dividing a Polynomial by a Monomial
(18 - 12) \div 3 can be done two ways: 6 \div 3 = 2, or \tfrac{18}{3} - \tfrac{12}{3} = 6 - 4 = 2. Likewise (ax + bx) \div x = \frac{ax}{x} + \frac{bx}{x} = a + b.
Rule. Divide each term of the polynomial by the monomial and write the partial quotients one after another.
A repeated parenthesis can be the "monomial" too. Treat it as one letter: \dfrac{3x(3x+4) - 4y(3x+4)}{3x+4} = 3x - 4y.
Quick check
\dfrac{12m^3 - 8m^2 + 4m}{4m}
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 each term by the monomial.
\dfrac{6x^2-4x}{2x}
\dfrac{9x-18x^4}{-3x}
\dfrac{4xy-12x^2}{-4x}
\dfrac{9ax^3-12x^5}{-3ax^2}
\dfrac{25x^2y+30xy^5}{-5xy}
\dfrac{16bx^4-36x^2}{4bx^2}
\dfrac{14x^3y^4-28x^5y^6}{7x^2y^3}
\dfrac{4x^4y-8x^6y^2+12x^3y^4}{4x^3y}
\dfrac{a^3cd^2-a^2c^3}{a^2cd}
\dfrac{ax^4-bx^3+cx^2}{-x^2}
\dfrac{15a^2b^2+9a^4b^2-30a^6b^2}{-3a^2b^2}
\dfrac{16a^4b^5c^6-24a^5b^6c^7-48a^6b^7c^8}{8a^3b^2c}
\dfrac{85xyz-51x^2yz^2+102x^3yz^3-170x^5y^5z}{-17xyz}
\dfrac{4(x-3)+a(x-3)}{x-3}
\dfrac{3x(3x+4)-4y(3x+4)}{3x+4}
\dfrac{5a(2x^2-y)-3b(2x^2-y)}{2x^2-y}
\dfrac{(a+b)^4-3(a+b)^3}{(a+b)^2}
\dfrac{21(x-y)^7-35(x-y)^5}{-7(x-y)^5}
\dfrac{16(3x-4)^4-24(3x-4)^5-48(3x-4)^7}{-8(3x-4)^4}
\dfrac{-5(ac^2-2d)^3+x(ac^2-2d)}{5(ac^2-2d)}
\dfrac{4x^4-8x^{3a}-6x^{2a-2}}{2x^3}
\dfrac{3x^{a}-2x^{a+1}-x^{a+2}+x^2}{x^2}
\dfrac{6x^{2a-3}-12x^{4a+4}-18x^{3a+5}}{-3x^{2a}}