1.11 Fractional and rational expressions

Many water-engineering rates and formulas compare one algebraic quantity with another through division.

Fractional expression. A quotient of two algebraic expressions, with all values that make the denominator zero excluded.

Rational expression. A fractional expression whose numerator and denominator are polynomials.

A fractional expression is a quotient of algebraic expressions. It is rational when its numerator and denominator are polynomials. Its domain excludes values that make an original denominator zero. State restrictions before cancelling factors:

\[\frac{x^2-16}{x-4}=x+4,\qquad x\ne4.\]

Worked Example: Simplifying while retaining the domain

For \((x^2-16)/(x-4)\), the original denominator requires \(x\ne4\). Factoring and cancelling gives

\[\frac{(x-4)(x+4)}{x-4}=x+4,\qquad x\ne4.\]

The restriction remains because the original expression is undefined at 4.

Try It. Simplify \((x^2-25)/(x-5)\) and state the restriction.
Check Your Work \(x+5\), with \(x\ne5\).

Four domain patterns recur. Polynomials allow all real inputs. Rational expressions exclude denominator zeros. An even root requires a nonnegative radicand. An even root in a denominator requires a strictly positive radicand. When several restrictions occur, their intersection is the domain.

Worked Example: Simplifying a hydraulic expression

For \(q\ne5\),

\[\frac{q^2-25}{q-5}=\frac{(q-5)(q+5)}{q-5}=q+5.\]

The simplified form still excludes \(q=5\) because the original expression is undefined there.

Try It. Simplify \((v^2-36)/(v-6)\) and retain the restriction.
Check Your Work \(v+6\), with \(v\ne6\).

Multiplication and division use fraction rules. Factor first and record restrictions before cancelling. When dividing, also exclude values that make the divisor zero before multiplying by its reciprocal.

Worked Example: Multiplying rational expressions

Simplify

\[\frac{3x^2+5x+2}{x-6}\cdot\frac{10x-5}{x+1}.\]

Factoring gives

\[\frac{(3x+2)(x+1)}{x-6}\cdot\frac{5(2x-1)}{x+1} =\frac{5(3x+2)(2x-1)}{x-6},\]

with \(x\ne6,-1\).

Try It. Simplify \([(x^2-4)/(x+3)]\,[3/(x-2)]\) and state the restrictions.
Check Your Work \(3(x+2)/(x+3)\), with \(x\ne-3,2\).

Addition and subtraction require a common denominator. Factor the denominators, construct the least common denominator, rewrite each numerator, combine, and then simplify.

Worked Example: Adding rational expressions

Using \(x(x+1)\) as the common denominator,

\[\frac2x+\frac3{x+1}=\frac{2(x+1)+3x}{x(x+1)}=\frac{5x+2}{x(x+1)},\]

where \(x\ne0,-1\).

Try It. Add \(1/x+2/(x-1)\).
Check Your Work \((3x-1)/[x(x-1)]\), with \(x\ne0,1\).

A compound fraction can be cleared by multiplying its entire numerator and denominator by the least common denominator of every smaller fraction. A denominator containing \(A+B\sqrt C\) can be rationalized using the conjugate \(A-B\sqrt C\), since their product is \(A^2-B^2C\).

1.11.1 A complete simplification routine

Factor first, state restrictions from every original denominator, perform the required operation, and cancel only common factors. When dividing, values that make the divisor zero must also be excluded. For addition and subtraction, build the least common denominator from each distinct factor at its greatest required power.

For compound fractions, multiply every term in the numerator and denominator by the same least common denominator. For a radical binomial, the conjugate changes the denominator into a difference of squares. These operations change form without changing value, provided all original restrictions are retained.

Since \(2-x=-(x-2)\), changing the direction of a factor introduces \(-1\). This is useful when constructing a common denominator, but omitting the negative sign changes the expression. Cancellation applies only to common factors in products. A term in a sum cannot be cancelled.

As an error check, evaluate the original and simplified forms at a convenient permitted value. Matching values do not prove equivalence, but different values immediately reveal an error.

Worked Example: Combining parallel flow times

If two identical-volume flow paths would individually require \(t_1\) and \(t_2\) hours, the combined reciprocal time is \(1/t_1+1/t_2\). For \(t_1=6\) and \(t_2=4\),

\[\frac16+\frac14=\frac2{12}+\frac3{12}=\frac5{12}.\]

The corresponding combined time is \(12/5=2.4\) h.

Try It. Find the combined time when \(1/t=1/3+1/6\).
Check Your Work \(1/t=1/2\), so \(t=2\) h.

Applied Problem: Combining two pump rates. Pump A can fill a tank in 8 h and Pump B can fill it in 12 h. When both pumps operate, their fractional tank-per-hour rates add. Form a rational expression for the combined rate and determine the filling time.

Check Your Work The combined rate is \(1/8+1/12=3/24+2/24=5/24\) tank per hour. Therefore, the filling time is the reciprocal, \(24/5=4.8\) h.

1.11.2 Practice Problems

  1. State the domain of \(1/(x+4)\).
    Check Your Work \(x\ne-4\).
  2. Simplify \((x^2-9)/(x+3)\).
    Check Your Work \(x-3\), with \(x\ne-3\).
  3. Multiply \(3/x\) and \(2/(x-1)\).
    Check Your Work \(6/[x(x-1)]\), with \(x\ne0,1\).
  4. Divide \(x/4\) by \(3x/8\).
    Check Your Work \(2/3\), with \(x\ne0\).
  5. Add \(1/x+1/(x+2)\).
    Check Your Work \(2(x+1)/[x(x+2)]\), with \(x\ne0,-2\).
  6. Simplify the hydraulic expression \((q^2-16)/(q-4)\) and state the restriction.
    Check Your Work \(q+4\), with \(q\ne4\).
  7. Two pumps have fractional rates \(1/10\) and \(1/15\) tank per hour. Find their combined rate and filling time.
    Check Your Work The rate is \(1/10+1/15=1/6\) tank per hour, so the time is 6 h.
  8. Simplify \((q^2-9)/(q+3)\cdot 2/(q-3)\) and state all restrictions.
    Check Your Work The expression simplifies to 2, with \(q\ne-3,3\).