Lesson 6 · from §65 of the 1910 text
Mixed Expressions
A mixed expression has a whole (integral) part and a fractional part, like x + 2 + \dfrac{3}{x + 1}.
In arithmetic, 4\tfrac23 means 4 + \tfrac23. In algebra, a\dfrac{b}{c} means a times \dfrac{b}{c}! So in algebra the two parts are always joined by a plus or minus sign.
If the numerator's degree is at least the denominator's, you can usually turn the fraction into a mixed expression by long division, exactly like \frac{14}{3} = 4 + \frac23. (Fractions like \frac{x^2}{a^2 + b^2} or \frac{xy}{x^2 + y^2} can't be reduced this way.)
Example. Reduce \dfrac{4x^4 - 6x^3 - x^2 - 1}{x^3}. Divide each term: 4x - 6 + \dfrac{-x^2 - 1}{x^3} = 4x - 6 - \dfrac{x^2 + 1}{x^3}.
Rule. Divide until the remainder has lower degree than the divisor. Write the remainder over the divisor and attach it to the quotient with a plus sign.
Going the other way, from a mixed expression to a single fraction, is what you did in Example 3 of adding fractions.
Quick check
\dfrac{x^2+3x+5}{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.
Reduce each fraction to a mixed expression: a whole part plus (or minus) a fraction. Type it like 3x - 2 + 2/(5x).
\dfrac{15x^2-10x+2}{5x}
\dfrac{24a^3-6a^2-14}{6a}
\dfrac{c}{c+1}
\dfrac{d^3+1}{d-1}
\dfrac{27x^3-y^3}{3x+y}
\dfrac{y^4+y^2+1}{y^2-y-1}
\dfrac{a^4+a^2b^2+b^4}{a^2+ab-b^2}
\dfrac{3y^3-11}{y+3}
\dfrac{(a+b)^3}{a^3+b^3}
\dfrac{16a^4+b^4}{2a-1}
\dfrac{5a^3+3a^2-6}{5a^2+3a+2}
\dfrac{x^6+y^6}{x^2-y^2}
\dfrac{x^4}{x+1}
\dfrac{(a^2+b)^2}{a^2+b^2}