Algebraic expressions
A workout app on your phone calculates calories burned with a formula like $C(t) = 0.15t^2 + 8t$, where $t$ is the number of minutes of exercise. That formula is an algebraic expression: a recipe written with letters that works for any $t$ without repeating the reasoning each time. On this page you will learn to read, write, and operate with them. How many calories does someone burn in 30 minutes? We will solve it at the end.
Algebraic language
Everyday language is ambiguous; mathematical language must be precise. An unknown or variable is a letter that represents a number we don't know yet or that can change. Translating a statement is done in parts: first each operation on a letter, then how they are combined.
| Everyday language | Expression |
|---|---|
| The sum of $a$ and $b$ | $a + b$ |
| The difference of $a$ and $b$ | $a - b$ |
| The product of $a$ and $b$ | $ab$ |
| The quotient (or ratio) of $a$ divided by $b$ | $\dfrac{a}{b}$ |
| Twice $x$ | $2x$ |
| The square of $x$ | $x^2$ |
| An even / odd number | $2n$ / $2n+1$ |
The remainder of dividing $a$ by $b$ can also be expressed with letters: $a = bq + r$, where $q$ is the quotient and $r$ the remainder.
Express the following statement in algebraic terms: 'The cube of a number minus the square of another, different number'.
Express the following statement in algebraic terms: 'The cube of a number minus the square of another, different number'.
Write the algebraic expression that represents the perimeter of a rectangle whose base is $3x$ and whose height is $x + 2$.
Write the algebraic expression that represents the perimeter of a rectangle whose base is $3x$ and whose height is $x + 2$.
Monomials: coefficient, literal part, and degree
Depending on the number of terms, an expression is a monomial (1 term), a binomial (2), or a trinomial (3); in general, a polynomial.
Identify the coefficient, the literal part and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.
Identify the coefficient, the literal part and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.
Polynomials: degree and numerical value
Not every expression with letters is a polynomial: the exponents must be natural numbers and the letter cannot be in a denominator or inside a root.
The numerical value of an expression is the result of substituting numbers for the letters and calculating, following the order of operations.
Calculate the numerical value of the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.
Calculate the numerical value of the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.
State which of the following algebraic expressions are polynomials and justify your answer: $A(x) = 3x^2 - \frac{5}{x}$ $B(x) = \sqrt{2}x^3 - x$ $C(x) = 4x^{1/2} + 1$
State which of the following algebraic expressions are polynomials and justify your answer:
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$A(x) = 3x^2 - \frac{5}{x}$
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$B(x) = \sqrt{2}x^3 - x$
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$C(x) = 4x^{1/2} + 1$
Operations with monomials
You can only add or subtract like monomials (same literal part), operating on their coefficients. To multiply or divide, the monomials do not need to be like: you operate on the coefficients on one side and on each letter on the other, adding or subtracting exponents.
Perform the following addition and subtraction of like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.
Perform the following addition and subtraction of like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.
Multiply the following monomials: $(-3a^2b^3) \times (4ab^4)$.
Multiply the following monomials: $(-3a^2b^3) \times (4ab^4)$.
Perform the following division of monomials and state the coefficient, literal part and degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.
Perform the following division of monomials and state the coefficient, literal part and degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.
Parameters and conditions
Sometimes the statement gives a numerical value of the polynomial and asks for an unknown letter (a parameter). You substitute, get an equation, and solve.
If there are two parameters, each condition gives an equation and you solve the system.
Determine the value of $k$ such that the numerical value of the polynomial $P(x) = 2x^3 - kx + 4$ at $x = 1$ is equal to $5$.
Determine the value of $k$ such that the numerical value of the polynomial $P(x) = 2x^3 - kx + 4$ at $x = 1$ is equal to $5$.
Find the values of the parameters $a$ and $b$ in the polynomial $P(x) = ax^2 + bx - 5$ knowing that its numerical value for $x = 1$ is $-2$ and for $x = -1$ is $-4$.
Find the values of the parameters $a$ and $b$ in the polynomial $P(x) = ax^2 + bx - 5$ knowing that its numerical value for $x = 1$ is $-2$ and for $x = -1$ is $-4$.
Addition, subtraction, and product of polynomials
To add or subtract polynomials, you combine like terms. When subtracting, the minus sign changes the sign of all the terms of the polynomial being subtracted.
To multiply, you apply the distributive property: each term of the first polynomial is multiplied by each term of the second, and then like terms are combined. It holds that $(a+b)(c+d) = ac + ad + bc + bd$.
An open-topped box is to be made from a rectangular cardboard sheet measuring $40$ cm long by $30$ cm wide. To do this, squares with side length $x$ are cut out from…
An open-topped box is to be made from a rectangular cardboard sheet measuring $40$ cm long by $30$ cm wide. To do this, squares with side length $x$ are cut out from each of the four corners and the flaps are folded upwards. Write the algebraic expression for the volume of the box in terms of $x$ and calculate this volume if $x = 5$ cm.
Factoring to simplify
Factoring is writing an expression as a product. It is used to simplify algebraic fractions: if the numerator and denominator share a factor, it cancels. These are the factorizations you need here:
The green area of a garden of $a$ by $b$ meters with two crossing paths of width $x$ is found by removing the paths: what remains is a rectangle of $(a-x)$ by $(b-x)$ meters, split into four parts.
Calculate the numerical value of the algebraic expression $P(x,y) = \frac{x^2 - 2xy + y^2}{x - y}$ for $x = \frac{1}{2}$ and $y = -\frac{3}{2}$. Hint: simplify the…
Calculate the numerical value of the algebraic expression $P(x,y) = \frac{x^2 - 2xy + y^2}{x - y}$ for $x = \frac{1}{2}$ and $y = -\frac{3}{2}$. Hint: simplify the expression before substituting the values.
Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, simplify it fully by carrying out the division and use the result to calculate its numerical…
Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, simplify it fully by carrying out the division and use the result to calculate its numerical value mentally when $a = 4.5$ and $b = 1.5$.
A rectangular garden has dimensions of $a$ meters long by $b$ meters wide. Two perpendicular paths of width $x$ are built parallel to the sides of the garden, forming a…
A rectangular garden has dimensions of $a$ meters long by $b$ meters wide. Two perpendicular paths of width $x$ are built parallel to the sides of the garden, forming a cross. Write the factored algebraic expression for the area of the remaining green area.
Fractions with letters and solving for a variable
Algebraic fractions are added just like numerical ones: you find equal denominators (the product of the denominators works) and add the numerators.
$$\frac{1}{R_1} + \frac{1}{R_2} = \frac{R_2}{R_1R_2} + \frac{R_1}{R_1R_2} = \frac{R_1 + R_2}{R_1R_2}$$To solve for a letter that is in the denominator, you take the reciprocal of both sides of the equation (if two fractions are equal, their reciprocals are also equal).
In an electrical circuit, the equivalent resistance $R_{eq}$ of three resistors $R_1$, $R_2$ and $R_3$ connected in parallel is obtained from the formula…
In an electrical circuit, the equivalent resistance $R_{eq}$ of three resistors $R_1$, $R_2$ and $R_3$ connected in parallel is obtained from the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Rearrange for $R_{eq}$ and write its algebraic expression as a single fraction in terms of $R_1$, $R_2$ and $R_3$.
Algebraic expressions in real-world problems
Perimeters, areas, volumes, rates, budgets, or recipes are written as algebraic expressions when some measurement is not fixed. Also remember: "a $20\,\%$ discount" means paying $80\,\%$, that is, multiplying by $0.8$.
A clothing store offers a discount of $20\%$ on all its items. If you buy $x$ shirts that originally cost $500$ pesos each and $y$ pairs of pants that cost $800$ pesos…
A clothing store offers a discount of $20\%$ on all its items. If you buy $x$ shirts that originally cost $500$ pesos each and $y$ pairs of pants that cost $800$ pesos each, write the algebraic expression for the total cost of the purchase. Calculate the cost if you buy $3$ shirts and $2$ pairs of pants.
The kinetic energy of a moving body is given by the expression $E_c = \frac{1}{2}mv^2$, where $m$ is the mass in kilograms and $v$ the speed in meters per second.…
The kinetic energy of a moving body is given by the expression $E_c = \frac{1}{2}mv^2$, where $m$ is the mass in kilograms and $v$ the speed in meters per second. Calculate the kinetic energy of a $800$ kg vehicle traveling at $20$ m/s.
The figure shows a square with side length $2x$ containing inside it a circle of radius $x$. Write the algebraic expression that represents the area of the shaded region…
The figure shows a square with side length $2x$ containing inside it a circle of radius $x$. Write the algebraic expression that represents the area of the shaded region (the region of the square outside the circle).
A phone company charges a fixed monthly fee of $300$ pesos, plus $2.40$ pesos for each minute of calls $m$ and $1$ peso for each megabyte consumed $d$. Write the…
A phone company charges a fixed monthly fee of $300$ pesos, plus $2.40$ pesos for each minute of calls $m$ and $1$ peso for each megabyte consumed $d$. Write the algebraic expression for the monthly bill and calculate its total for a month in which $100$ minutes and $500$ megabytes have been used.
A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions representing its volume and its curved surface area in terms of $x$.
A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions representing its volume and its curved surface area in terms of $x$.
More elaborate models
With practice, expressions become richer: rates that are compared, general sums, plots of land with constraints, and proofs with letters.
One last tip for fenced plots of land: if a river covers one side, the fence only goes around three. With width $x$ and $L$ meters of fence, the two width sides use $2x$ and the length measures $L - 2x$.
Prove algebraically that the difference between the squares of two consecutive odd numbers is always a multiple of $8$. (Remember that an odd number can be written in…
Prove algebraically that the difference between the squares of two consecutive odd numbers is always a multiple of $8$. (Remember that an odd number can be written in the form $2n+1$).
Two car rental agencies offer different rates. Agency A charges a fixed fee of $600$ pesos plus $4$ pesos per kilometer traveled. Agency B charges a fixed $1\,000$ pesos…
Two car rental agencies offer different rates. Agency A charges a fixed fee of $600$ pesos plus $4$ pesos per kilometer traveled. Agency B charges a fixed $1\,000$ pesos plus $3$ pesos per kilometer. Write the algebraic expression representing the difference in cost between Agency A and Agency B for a trip of $k$ kilometers, and determine for how many kilometers the cost is exactly the same.
Write the algebraic expression that gives the sum of the first $n$ positive integer multiples of $3$, knowing that the sum of the first $n$ natural numbers is…
Write the algebraic expression that gives the sum of the first $n$ positive integer multiples of $3$, knowing that the sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$. Use this expression to calculate the sum of the first $50$ multiples of $3$.
A farmer has $L$ meters of fencing to enclose a rectangular plot of land adjacent to a straight river. Since the river acts as a natural barrier, no fencing is needed…
A farmer has $L$ meters of fencing to enclose a rectangular plot of land adjacent to a straight river. Since the river acts as a natural barrier, no fencing is needed along that side. If we call $x$ the width of the plot (the dimension perpendicular to the river), write the algebraic expression for the area of the enclosure in terms of $L$ and $x$. What is the area if he has $100$ meters of fencing and the width is $20$ meters?
Back to the beginning: for $t = 30$ minutes, $C(30) = 0.15(30)^2 + 8(30) = 0.15 \times 900 + 240 = 135 + 240 = 375$ calories.