Algebraic Expressions
A fitness app estimates the calories you burn with a formula like $C(t) = 0.15t^2 + 8t$, where $t$ is the number of minutes you work out. That formula is an algebraic expression: a recipe written with letters that works for any $t$, so you don't have to redo the reasoning every time. In this section you'll learn the vocabulary of these expressions, how to simplify and evaluate them, and how to write them yourself from a word problem or a picture.
Monomials: coefficient, variable part, and degree
Take the monomial below. You read it in three pieces.
$$M(x,y) = -\frac{2}{3}x^4y^2$$Identify the coefficient, the variable part, and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.
Identify the coefficient, the variable part, and the degree of the following monomial: $M(x,y) = -\frac{2}{3}x^4y^2$.
Polynomials: what they are and their degree
Not every expression with letters is a polynomial. The exponents on the variables must be whole numbers ($0, 1, 2, \dots$), and a variable can't appear in a denominator or inside a radical. The coefficients, however, can be any real numbers, such as $\frac{1}{2}$ or $\sqrt{2}$.
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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$C(x) = 4x^{1/2} + 1$
Evaluating expressions and finding unknown coefficients
To evaluate an expression, substitute a number for each variable and then follow the order of operations. Powers come before multiplication, and multiplication comes before adding and subtracting.
Here is the graph of the same polynomial. The value $P(-2) = 24$ is the height of the graph above $x = -2$.
Sometimes a problem gives you the value of a polynomial and asks for an unknown coefficient. Substitute the number, and you get an equation to solve.
When there are two unknowns, you need two pieces of information. That gives two equations.
Evaluate the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.
Evaluate the polynomial $P(x) = 3x^2 - 5x + 2$ for $x = -2$.
Determine the value of $k$ so that the value of the polynomial $P(x) = 2x^3 - kx + 4$ at $x = 1$ is equal to $5$.
Determine the value of $k$ so that the 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 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 value for $x = 1$ is $-2$ and for $x = -1$ is $-4$.
Operations with monomials
You can add or subtract monomials only when they are like terms (same variable part). Then you add or subtract the coefficients and keep the variable part. To multiply or divide monomials they do not have to be like terms. Work with the coefficients on one side and with each variable on the other, adding or subtracting exponents.
Combine the following like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.
Combine the following like terms and simplify the result: $3x^2y - 5x^2y + 8x^2y$.
Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.
Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.
Perform the following division of monomials and indicate the coefficient, the variable part, and the degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.
Perform the following division of monomials and indicate the coefficient, the variable part, and the degree of the resulting monomial: $\frac{-24x^5y^4z}{6x^2y^3}$.
From words to algebra
To translate a sentence, go piece by piece. First write what happens to each variable, then decide how the pieces are combined.
Here are some translations you will use again and again.
| Words | Algebra |
|---|---|
| twice a number | $2n$ |
| 5 more than a number | $n + 5$ |
| 5 less than a number | $n - 5$ |
| the square of the sum of $a$ and $b$ | $(a+b)^2$ |
| the sum of the squares of $a$ and $b$ | $a^2 + b^2$ |
| twice the square of $x$ | $2x^2$ |
| the square of twice $x$ | $(2x)^2$ |
Real situations often involve a fixed amount plus a rate. For example, a phone plan costs a flat fee plus a charge per unit. The fixed amount is added once, and each rate is multiplied by how much you use.
Write the following statement as an algebraic expression: "The cube of a number minus the square of a different number."
Write the following statement as an algebraic expression: "The cube of a number minus the square of a different number."
Write the algebraic expression that represents the perimeter of a rectangle whose base measures $3x$ and whose height measures $x + 2$.
Write the algebraic expression that represents the perimeter of a rectangle whose base measures $3x$ and whose height measures $x + 2$.
A phone company charges a fixed monthly fee of $\$15$, plus $\$0.12$ for each minute of call $m$ and $\$0.05$ for each megabyte consumed $d…
A phone company charges a fixed monthly fee of $\$15$, plus $\$0.12$ for each minute of call $m$ and $\$0.05$ for each megabyte consumed $d$. Write the algebraic expression for the monthly bill and calculate the total for a month in which $100$ minutes and $500$ megabytes have been used.
Formulas from money and science
Prices, rates, and physics formulas are algebraic expressions in disguise. To write one, ask yourself: what changes (that's the variable), and what stays fixed (that's a number)?
When something is on sale, the discount applies to the whole total. A $20\%$ discount means you pay $80\%$ of the price, so you multiply the total by $0.8$.
Many science formulas work the same way. Once you know the numbers, you just evaluate. For example, the area of a circle is $A = \pi r^2$. With $r = 3$ ft, $A = \pi \cdot 3^2 = 9\pi \approx 28.3$ ft².
A rate is a good place to see why expressions are useful. Suppose Plan A costs $20 + 0.10x$ dollars and Plan B costs $5 + 0.25x$ dollars for $x$ units. You can compare the plans by graphing both expressions. Where the lines cross, the costs are equal.
A clothing store offers a $20\%$ discount on all its items. If you buy $x$ shirts that originally cost $\$25$ each and $y$ pairs of pants that cost $\$40$ each, write…
A clothing store offers a $20\%$ discount on all its items. If you buy $x$ shirts that originally cost $\$25$ each and $y$ pairs of pants that cost $\$40$ 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 pounds and $v$ is the speed in feet 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 pounds and $v$ is the speed in feet per second. Calculate the kinetic energy of a vehicle of $800$ lb traveling at $20$ ft/s.
Two car rental agencies offer different rates. Agency A charges a fixed fee of $\$30$ plus $\$0.20$ per mile driven. Agency B charges a fixed $\$50$ plus $\$0.15$ per…
Two car rental agencies offer different rates. Agency A charges a fixed fee of $\$30$ plus $\$0.20$ per mile driven. Agency B charges a fixed $\$50$ plus $\$0.15$ per mile. Write the algebraic expression that represents the difference in cost between Agency A and Agency B for a trip of $k$ miles, and determine for how many miles the cost is exactly the same.
Geometry: areas, volumes, and surfaces
Perimeters, areas, and volumes are natural algebraic expressions when a length isn't fixed yet. For a shaded region, ask: what is the big shape, and what do I take away?
Here the whole square has area $(2x)^2 = 4x^2$, and the circle has area $\pi x^2$. The shaded part is what's left over: $4x^2 - \pi x^2$.
The figure shows a square of side $2x$ containing a circle of radius $x$ inside it. Write the algebraic expression that represents the area of the shaded region (the…
The figure shows a square of side $2x$ containing a circle of radius $x$ inside it. Write the algebraic expression that represents the area of the shaded region (the area of the square outside the circle).
A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions that represent its volume and its lateral surface area in terms of…
A cylindrical tank has a base radius $r = x$ and a height $h = 3x$. Write the algebraic expressions that represent its volume and its lateral surface area in terms of $x$.
A rectangular garden has dimensions of $a$ feet long by $b$ feet wide. Two perpendicular paths of width $x$ are built parallel to the sides of the garden, forming a…
A rectangular garden has dimensions of $a$ feet long by $b$ feet 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 zone.
Building a model from a situation
Harder problems give you a story and ask you to write one expression that depends on a single variable. The method is always the same.
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Draw a picture and label it.
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Choose a variable for the quantity that can change.
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Write every other length or amount in terms of that variable.
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Put them together with a formula.
A box without a lid is to be made from a rectangular sheet of cardboard $40$ in long by $30$ in wide. To do this, squares of side $x$ are cut out from each of the four…
A box without a lid is to be made from a rectangular sheet of cardboard $40$ in long by $30$ in wide. To do this, squares of side $x$ are cut out from each of the four corners and the flaps are folded up. Write the algebraic expression for the volume of the box in terms of $x$ and calculate this volume if $x = 5$ in.
A farmer has $L$ feet of fencing to enclose a rectangular plot of land adjacent to a straight river. As the river acts as a natural barrier, they do not need to put a…
A farmer has $L$ feet of fencing to enclose a rectangular plot of land adjacent to a straight river. As the river acts as a natural barrier, they do not need to put a fence on 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 they have $100$ feet of fencing and the width is $20$ feet?
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 $3$.
Simplify before you substitute
Sometimes an expression looks messy but collapses to something simple. Spotting a pattern first can save a lot of arithmetic, and you can verify the pattern by multiplying it out.
Algebraic expressions can also be fractions. To add fractions with letters, use a common denominator, exactly as with numbers.
Evaluate 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…
Evaluate 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 $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, fully simplify it by carrying out the division and use the result to mentally calculate its…
Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, fully simplify it by carrying out the division and use the result to mentally calculate its value when $a = 4.5$ and $b = 1.5$.
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 $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$. Solve for $R_{eq}$ and write its algebraic expression as a single fraction in terms of $R_1$, $R_2$, and $R_3$.
Proving with algebra
Testing a few numbers can suggest that a statement is true, but it can't prove it. Algebra lets you prove a statement for every number at once. You write the numbers as an expression, simplify, and show the result has the property you want.
Useful ways to write numbers:
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An even number: $2n$
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An odd number: $2n + 1$
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The next odd number after $2n+1$: $2n + 3$ (consecutive odd numbers differ by $2$)
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A multiple of $8$: $8k$ for some integer $k$
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 expressed 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 expressed in the form $2n+1$).
Back to the start: If you work out for $30$ minutes, the app finds your calories by evaluating at $t = 30$: $C(30) = 0.15\cdot 30^2 + 8\cdot 30 = 0.15\cdot 900 + 240 = 135 + 240 = 375$ calories.