Lesson 3 · from §70 of the 1910 text
Polynomial Denominators
Polynomial denominators
When the denominators are polynomials, factor them, find the L.C.M., and multiply through just as before.
Example 1. Solve \dfrac{x^2}{2x^3 - 2} = \dfrac{2x}{3x^2 + 3x + 3} - \dfrac{1}{6x - 6}.
Factor the denominators: 2(x - 1)(x^2 + x + 1), 3(x^2 + x + 1), and 6(x - 1). The L.C.M. is 6(x - 1)(x^2 + x + 1). Multiplying each term by it and canceling, 3x^2 = 4x(x - 1) - (x^2 + x + 1). 3x^2 = 4x^2 - 4x - x^2 - x - 1. So 5x = -1 and x = -\frac15.
Roots that aren't roots
Multiplying by an expression that contains x can bring in a "root" that doesn't satisfy the original equation: one that makes a denominator zero. Always check.
Example 2. Solve \dfrac{3x - 2}{x - 2} = \dfrac{4x - 4}{x - 2} + 1.
Multiplying by x - 2: 3x - 2 = 4x - 4 + x - 2, so -2x = -4 and x = 2. But x = 2 makes the denominators zero, and division by zero means nothing. The equation has no root. The book calls such an equation impossible. Type none for these.
Rule. Clear of fractions using the L.C.M. of the denominators, solve, and then reject any value that makes a denominator zero.
Sometimes clearing fractions leaves a quadratic. Solve it by factoring and check both roots.
Try one: solve \dfrac{3}{x - 1} = \dfrac{2}{x + 1}.
Quick check
\displaystyle \frac{3}{x-1}=\frac{2}{x+1}
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.
Solve and check. A number that makes a denominator zero is not a root. If an equation has no root, type none. If it has two roots, separate them with a comma.
\displaystyle \frac{32}{x}=5
\displaystyle \frac{25}{3x}=5
\displaystyle 5x+\frac{x-2}{4}=4x+7
\displaystyle 3x-\frac{x-1}{4}-\frac{8x}{5}=6
\displaystyle \frac{4}{x}-\frac{12+x}{3x}=\frac{4}{3}
\displaystyle \frac{5}{6}-\frac{3x-5}{4x}+\frac{x-2}{3x}=0
\displaystyle \frac{x+5}{5x}-\frac{3(x+1)}{x}=3\tfrac{1}{5}
\displaystyle 5x-4x\left(3-\frac{2}{x}\right)+\frac{1}{4}=3
\displaystyle x-\frac{3x}{5}\left(\frac{10}{x}-4\right)+3\tfrac{3}{5}=18
\displaystyle \frac{8x-7}{4x}+3\tfrac{1}{2}=\frac{\frac{3}{2}-x}{x}
\displaystyle \frac{2x-3}{3+2x}=4
\displaystyle \frac{x-2}{x-3}=\frac{15}{16}
\displaystyle \frac{5}{7x+5}-\frac{1}{8}=0
\displaystyle \frac{3x^2-7x-4}{4x^2-10x-8}=\frac{3}{4}
\displaystyle \frac{1}{x-2}=\frac{3}{x-3}
\displaystyle \frac{1}{x-3}=\frac{x-6}{x+3}+\frac{x}{x+3}
\displaystyle \frac{x-3}{x+4}=\frac{x-9}{x+5}
\displaystyle \frac{x-2}{x+2}+\frac{4}{x+2}+2=0
\displaystyle \frac{3x}{4}-\frac{2}{x-2}=\frac{3x-2}{4}
\displaystyle \frac{4}{x-3}+\frac{3x+4}{6}=\frac{x}{2}
\displaystyle \frac{2x+3}{x-5}+4=\frac{x+8}{x-5}
\displaystyle \frac{x}{4}-\frac{5}{4x-12}=\frac{2x+\frac{5}{4}}{3}
\displaystyle \frac{x-4}{x+5}+\frac{7}{5}=\frac{3}{x+5}
\displaystyle \frac{x+4}{15x-5}-\frac{x}{5}=\frac{2x+\frac{20}{3}}{10}
\displaystyle \frac{4}{3x+6}+\frac{5}{7x+14}+43=0
\displaystyle \frac{7}{4x-12}+\frac{47}{220}=-\frac{3}{5x-15}
\displaystyle \frac{4x}{x+3}-\frac{6}{2x+6}=\frac{10x+11}{3x+9}
\displaystyle \frac{3}{x-2}=\frac{5}{x^2-25}+\frac{3x}{x^2-25}
\displaystyle \frac{1}{x-3}+\frac{2}{x+3}=-\frac{3}{x^2-9}
\displaystyle \frac{x+2}{x-2}=\frac{10-x^2}{4-x^2}-\frac{10}{x^2-4}
\displaystyle \frac{x-4}{x-5}+\frac{x-15}{x+4}=\frac{2x^2-10x-1}{x^2-x-20}
\displaystyle \frac{x+2}{x+3}+\frac{x+3}{x+2}=\frac{4x+9}{x^2+5x+6}