Operations with Algebraic Fractions
A tank is filled with two hoses: one takes $x$ hours by itself and the other takes $x+3$ hours. What part of the tank do they fill together in one hour? The answer is a sum of algebraic fractions, and it is handled just like $\dfrac{1}{4}+\dfrac{1}{6}$: with a common denominator. What changes is that, instead of prime factors, you are going to factor polynomials. At the end we will solve this problem.
Simplifying by Factoring the Numerator and Denominator
Simplifying an algebraic fraction means canceling the factors that the numerator and denominator have in common. The essential step is to factor both first: without factoring there is nothing to cancel, just as you can't simplify $\dfrac{10}{15}$ without seeing that both are multiples of $5$.
Notice the condition $x\neq-3$: the original fraction does not exist when the denominator equals $0$, and canceling does not change that.
Simplify the following algebraic fraction as far as possible by factorising the numerator and the denominator: $\frac{x^2 - 9}{x^2 + 6x + 9}$
Simplify the following algebraic fraction as far as possible by factorising the numerator and the denominator:
$\frac{x^2 - 9}{x^2 + 6x + 9}$
Multiplication and Division
Just like with numerical fractions: to multiply, multiply numerator by numerator and denominator by denominator; to divide, multiply by the reciprocal fraction. Factoring before multiplying lets you cancel as you go, without expanding large products.
This is useful for areas. If the solar panel on a rooftop has base $b=\dfrac{x^2-4}{x+1}$ m and height $h=\dfrac{x+1}{x-2}$ m, its area is $b\cdot h=\dfrac{(x-2)(x+2)}{x+1}\cdot\dfrac{x+1}{x-2}=x+2$ m².
Perform the following multiplication of algebraic fractions and express the result in its simplest form: $\frac{2x}{x-1} \times \frac{x^2 - 1}{4x^2}$
Perform the following multiplication of algebraic fractions and express the result in its simplest form:
$\frac{2x}{x-1} \times \frac{x^2 - 1}{4x^2}$
Divide the following algebraic fractions and simplify the result obtained: $\frac{x^2 + 2x}{x - 3} : \frac{x^2 - 4}{x^2 - 9}$
Divide the following algebraic fractions and simplify the result obtained:
$\frac{x^2 + 2x}{x - 3} : \frac{x^2 - 4}{x^2 - 9}$
The base of a rectangular solar panel is given by the expression $b = \frac{x^3 - x}{x^2 + 2x + 1}$ meters, and its height by $h = \frac{x^2 + x}{x^2 - 2x + 1}$ meters.…
The base of a rectangular solar panel is given by the expression $b = \frac{x^3 - x}{x^2 + 2x + 1}$ meters, and its height by $h = \frac{x^2 + x}{x^2 - 2x + 1}$ meters. Calculate the area of the solar panel, expressing the result as a single simplified algebraic fraction.
Addition and Subtraction: Least Common Multiple
With the same denominator, you add the numerators and the denominator does not change. With different denominators you need a common denominator: factor each denominator and take the least common multiple (LCM), that is, each distinct factor with its highest exponent.
When the denominators aren't so easy
If a denominator has degree $3$, look for an integer root using synthetic division and factor what remains. For example, $x^3-2x^2-x+2$ has root $x=1$:
| 1 | −2 | −1 | 2 | |
|---|---|---|---|---|
| 1 | 1 | −1 | −2 | |
| 1 | −1 | −2 | 0 |
The remainder is $0$ and the quotient is $x^2-x-2=(x-2)(x+1)$. Therefore $x^3-2x^2-x+2=(x-1)(x-2)(x+1)$.
Add the following algebraic fractions with common denominators: $\frac{3x - 1}{x^2 + 1} + \frac{x + 5}{x^2 + 1}$
Add the following algebraic fractions with common denominators:
$\frac{3x - 1}{x^2 + 1} + \frac{x + 5}{x^2 + 1}$
Subtract the following algebraic fractions by first finding a common denominator: $\frac{5}{x+2} - \frac{3}{x^2 - 4}$
Subtract the following algebraic fractions by first finding a common denominator:
$\frac{5}{x+2} - \frac{3}{x^2 - 4}$
Add the following algebraic fractions with linear denominators: $\frac{x}{x-2} + \frac{2}{x+3}$
Add the following algebraic fractions with linear denominators:
$\frac{x}{x-2} + \frac{2}{x+3}$
Perform the addition and subtraction of the following algebraic fractions. Use Ruffini's rule and algebraic identities to factorise the denominators first and find the…
Perform the addition and subtraction of the following algebraic fractions. Use Ruffini's rule and algebraic identities to factorise the denominators first and find the lowest common multiple:
$\frac{2x}{x^3 - 3x^2 + 3x - 1} - \frac{1}{x^2 - 2x + 1} + \frac{3}{x-1}$
Combined Operations: The Order of Operations
When sums, differences, products, and parentheses are mixed, you follow the usual order of operations: first parentheses, then multiplication and division (left to right), and finally addition and subtraction.
Evaluate the expression inside the brackets and then multiply to simplify the final result: $\frac{x^2 - x}{x+1} \times \left( \frac{1}{x} + \frac{1}{x-1} \right)$
Evaluate the expression inside the brackets and then multiply to simplify the final result:
$\frac{x^2 - x}{x+1} \times \left( \frac{1}{x} + \frac{1}{x-1} \right)$
Carry out the following combined operation of algebraic fractions and simplify as far as possible: $\frac{x^2+x}{x^2-1} - \frac{1}{x-1}$
Carry out the following combined operation of algebraic fractions and simplify as far as possible:
$\frac{x^2+x}{x^2-1} - \frac{1}{x-1}$
Perform the following combined operations respecting the order of operations and simplify the result as far as possible:…
Perform the following combined operations respecting the order of operations and simplify the result as far as possible:
$\left( \frac{x+1}{x-1} - \frac{x-1}{x+1} \right) : \frac{4x}{x^2 - 1}$
Complex Fractions
A complex fraction has fractions in its numerator, in its denominator, or in both. There are two approaches: reduce the top and bottom to a single fraction and divide, or multiply the numerator and denominator by the LCM of the inner denominators.
Simplify the following complex algebraic fraction by first simplifying the numerator and denominator independently: $\frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x}}$
Simplify the following complex algebraic fraction by first simplifying the numerator and denominator independently:
$\frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x}}$
Sums of Reciprocals
Sometimes a quantity is defined by a sum of reciprocals: $\dfrac{1}{T}=\dfrac{1}{A}+\dfrac{1}{B}$ (this is how, for example, teamwork times or the distances in the lens formula are combined). To find $T$ you add, simplify, and at the end flip the whole fraction.
When one sum appears inside another, work from the inside out. For $x+\dfrac{1}{\frac1x+\frac1{x+1}}$:
The thin lens formula in optics states that $\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}$. If the distance from the object to the optical center is…
The thin lens formula in optics states that $\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}$. If the distance from the object to the optical center is $s_o = \frac{x^2+x}{2}$ and the image distance is $s_i = \frac{x^2-1}{3}$, determine the simplified algebraic expression for the focal length $f$.
Work, Costs, and Mixtures
Go back to the tank from the start. If one hose fills it in $x$ hours, in one hour it fills $\dfrac1x$ of the tank (its work rate). The other fills $\dfrac{1}{x+3}$. Together:
$$\frac1x+\frac1{x+3}=\frac{(x+3)+x}{x(x+3)}=\frac{2x+3}{x(x+3)}$$That is the part of the tank they fill together in one hour. For example, with $x=3$: $\dfrac{9}{18}=\dfrac12$ of the tank per hour (they take 3 and 6 hours separately, and together $2$ hours). This closes the opening problem.
Two industrial 3D printers manufacture a batch of parts working together. The first printer takes $x$ hours to complete the job alone, and the second takes $x+3$ hours.…
Two industrial 3D printers manufacture a batch of parts working together. The first printer takes $x$ hours to complete the job alone, and the second takes $x+3$ hours. The fraction of the work they complete together in one hour is the sum of their individual work rates. Write down and simplify the algebraic fraction representing that fraction of work per hour.
The average production cost of an electronic component is given by the expression $C_m(x) = \frac{x^3 + 2x^2 - 5x - 6}{x^2 - 4}$, where $x$ is the production volume in…
The average production cost of an electronic component is given by the expression $C_m(x) = \frac{x^3 + 2x^2 - 5x - 6}{x^2 - 4}$, where $x$ is the production volume in thousands of units. Due to a subsidy adjustment, a fee represented by the fraction $\frac{x^2-1}{x+2}$ must be subtracted. Calculate the new expression for the average cost by first simplifying the first fraction.
In a chemistry laboratory, there are two solutions of the same acid. The first has a concentration of $C_1 = \frac{x}{x^2+2x+1}$ and the second $C_2 = \frac{2}{x^2-1}$.…
In a chemistry laboratory, there are two solutions of the same acid. The first has a concentration of $C_1 = \frac{x}{x^2+2x+1}$ and the second $C_2 = \frac{2}{x^2-1}$. Equal volumes of both solutions are mixed, so the concentration of the resulting mixture is the arithmetic mean of both concentrations, that is, $C_m = \frac{1}{2}(C_1 + C_2)$. Obtain the simplified algebraic expression for the final concentration of the mixture $C_m$.
Proving Identities with Fractions
To prove an equality, you work on only one side (usually the more complicated one) until you reach the other. You may factor, add, and simplify, but you may not move terms from one side to the other.
These factorizations are very useful:
If factors such as $(y-x)$ and $(x-y)$ appear, use the fact that $y-x=-(x-y)$. For example:
$$\frac{1}{x-y}+\frac{1}{y-x}=\frac{1}{x-y}-\frac{1}{x-y}=0$$With three variables, the trick is to bring everything to the same denominator, for example $(x-y)(x-z)(y-z)$, being careful with the signs of each term.
Show that the following algebraic equality holds for any values of the variables $x$ and $y$ that do not make the denominators zero, by carrying out the operations and…
Show that the following algebraic equality holds for any values of the variables $x$ and $y$ that do not make the denominators zero, by carrying out the operations and simplifying the left-hand side:
$\left( \frac{x^2 - y^2}{x+y} \right) \times \left( \frac{x^3 + y^3}{x^2 - xy + y^2} \right) = x^2 - y^2$
A motorboat travels along a canal from a jetty to a buoy at a speed $v+w$ (where $w$ is the downstream current speed) and returns at a speed $v-w$ (upstream). The…
A motorboat travels along a canal from a jetty to a buoy at a speed $v+w$ (where $w$ is the downstream current speed) and returns at a speed $v-w$ (upstream). The distance between the points is $d$. Obtain the simplified expression for the average speed of the round trip, given by $v_m = \frac{2d}{\frac{d}{v+w} + \frac{d}{v-w}}$, and check whether it is equivalent to the expression $\frac{v^2 - w^2}{v}$.
Carry out the indicated operations involving three variables and prove that the final result simplifies to a constant integer:…
Carry out the indicated operations involving three variables and prove that the final result simplifies to a constant integer:
$\frac{x^2}{(x-y)(x-z)} + \frac{y^2}{(y-x)(y-z)} + \frac{z^2}{(z-x)(z-y)}$
Partial Fraction Decomposition
Sometimes you have to do the opposite of adding: write a fraction as a sum of fractions with simpler denominators. You set up the decomposition with letters and find their values by equating numerators.
When a quadratic factor that can't be factored appears, such as $x^2+1$, its numerator has the form $Bx+C$. You expand and equate the coefficients of $x^2$, $x$, and the constant term.
Determine the numerical values of the parameters $A$, $B$ and $C$ such that the following partial fraction decomposition identity holds for all $x$:…
Determine the numerical values of the parameters $A$, $B$ and $C$ such that the following partial fraction decomposition identity holds for all $x$:
$\frac{5x^2 - x + 2}{x^3 - x^2 + x - 1} = \frac{A}{x-1} + \frac{Bx+C}{x^2+1}$
The Difference Quotient
In calculus, how fast a function changes is studied with $\dfrac{f(x+h)-f(x)}{h}$ (the average rate of change, the basis of the four-step rule for the derivative). Here the important part is the algebra: subtracting fractions and canceling the $h$ in the denominator, which will appear as a factor of the numerator.
If the function has several terms, work on each one separately: the polynomial part, such as $x^2$, uses $(x+h)^2=x^2+2xh+h^2$, and the fractional part is subtracted as above. Then you put them together.
The surface area of a cylindrical container designed to minimise the use of plastic material is given by the expression $S(x) = 2\pi x^2 + \frac{1000}{x}$. To analyse…
The surface area of a cylindrical container designed to minimise the use of plastic material is given by the expression $S(x) = 2\pi x^2 + \frac{1000}{x}$. To analyse the rate of change of material with respect to the radius $x$, an engineer needs to calculate and simplify the expression for the difference quotient $\frac{S(x+h) - S(x)}{h}$. Perform this operation and express the result as a single simplified algebraic fraction in terms of $x$ and $h$.