Algebraic Expressions
Your streaming subscription costs $x$ dollars a month. You split it evenly with two friends, so each person pays $\dfrac{x}{3}$. You don't know the exact price yet, but you already know the calculation to do. That is algebra: a rule that works for any number.
Writing Expressions from Words
Each key word turns into an operation. When a phrase has several steps, translate from the inside out, like following a recipe.
| In words | Algebraic expression |
|---|---|
| Twice a number, plus 5 | $2x + 5$ |
| Twice the sum of a number and 5 | $2(x + 5)$ |
| Half of a number, minus 3 | $\dfrac{x}{2} - 3$ |
| A \$5 coupon off the price $x$ | $x - 5$ |
| 5 less than a number | $x - 5$ |
In an online video game store, the price of a monthly subscription is $\$x…
In an online video game store, the price of a monthly subscription is $\$x$. Translate the following situations into algebraic language:
- a) The cost of three monthly subscriptions.
- b) The price of a subscription after applying a discount coupon of $\$5$.
- c) What a group of friends pays for a monthly subscription plus a game of $\$25$.
- d) Double the price of a subscription increased by $\$10$.
In a comic book shop, the price of a monthly subscription is $\$x$. For each additional comic you buy, they charge you $\$y$, but if you buy more than five, they apply a…
In a comic book shop, the price of a monthly subscription is $\$x$. For each additional comic you buy, they charge you $\$y$, but if you buy more than five, they apply a discount of $\$3$ to the total. Write the algebraic expression that represents the monthly cost of a customer who has bought $7$ additional comics.
Evaluating Expressions
Why parentheses? A variable is a placeholder for a whole number, sign included. Parentheses keep that number together.
An expression can have more than one variable, like $P(x, y) = 2xy + 1$. Substitute both. For $x = 3$ and $y = -1$: $2(3)(-1) + 1 = -6 + 1 = -5$.
Evaluating also lets you find a missing number. A value of $x$ that makes a polynomial equal $0$ is called a zero (or root) of the polynomial.
Evaluate the following algebraic expressions for the indicated values: a) $5x - 3$ for $x = 4$ b) $x^{2} + 2x + 1$ for $x = -2$ c) $2(a + b)$ for $a = 3$ and $b = 5$ d)…
Evaluate the following algebraic expressions for the indicated values:
- a) $5x - 3$ for $x = 4$
- b) $x^{2} + 2x + 1$ for $x = -2$
- c) $2(a + b)$ for $a = 3$ and $b = 5$
- d) $\frac{10 - x}{2}$ for $x = 6$
Find the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy + 2$ for the indicated values of the variables: a) $x = 2, y = -1$ b) $x = 0, y = 10$ c)…
Find the numerical value of the polynomial $P(x, y) = 3x^2y - 5xy + 2$ for the indicated values of the variables:
- a) $x = 2, y = -1$
- b) $x = 0, y = 10$
- c) $x = -2, y = -2$
Consider the polynomial $P(x) = 2x^3 - kx^2 + 4x - 12$. 1. Determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is…
Consider the polynomial $P(x) = 2x^3 - kx^2 + 4x - 12$.
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Determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is equal to zero ($x = 2$ is a root of the polynomial).
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Once $k$ is found, calculate the numerical value of $P(x)$ for $x = -1$.
Terms, Degree, and Like Terms
A term is a number, a variable, or numbers and variables multiplied together, like $7x^2$ or $-5ab^3$. A one-term expression is a monomial. Every monomial has three parts you should be able to name:
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The coefficient is the number multiplied by the variables. For $x^4y$ it is $1$ (an invisible $1$), and for $\frac{3}{4}z^2$ it is $\frac{3}{4}$.
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The variable part is the letters with their exponents.
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The degree of a monomial is the sum of the exponents of its variables. A plain number like $7$ has degree $0$.
An expression with several terms is a polynomial. It has a special name when it has two terms (binomial) or three terms (trinomial).
Combining like terms
Like terms have exactly the same variable part (same letters, same exponents). Only the coefficients can differ. You can add or subtract like terms by adding or subtracting their coefficients.
| Like terms | Not like terms |
|---|---|
| $3x$ and $-5x$ | $3x$ and $3x^2$ |
| $4ab$ and $ab$ | $4ab$ and $4a$ |
| $7$ and $-2$ | $7$ and $7x$ |
Why does this work? $3x + 5x$ means "3 of something plus 5 of the same thing", which is $8$ of them: $8x$. But $3x + 5x^2$ are different things, like 3 apples and 5 oranges.
Identify the elements of the following monomials by completing the information for each one: a) $7x^{2}$ b) $-5ab^{3}$ c) $x^{4}y$ d) $\frac{3}{4}z^{2}$ For each one,…
Identify the elements of the following monomials by completing the information for each one:
- a) $7x^{2}$
- b) $-5ab^{3}$
- c) $x^{4}y$
- d) $\frac{3}{4}z^{2}$
For each one, state: Coefficient, Variable part and Degree.
Classify the following algebraic expressions, indicating whether they are monomials, binomials or trinomials. In the case of polynomials (binomials and trinomials), also…
Classify the following algebraic expressions, indicating whether they are monomials, binomials or trinomials. In the case of polynomials (binomials and trinomials), also state their degree.
- a) $4x^{3} - 5x + 2$
- b) $-7ab^{2}$
- c) $x^{2} + 10$
- d) $3y^{4} - 2y^{2}$
Reduce the following algebraic expressions by adding or subtracting like terms: a) $3x + 5x - 2x$ b) $7a^{2} - 3a + 2a^{2} + 5a$ c) $4xy + 2x - xy + 3x$ d)…
Reduce the following algebraic expressions by adding or subtracting like terms:
- a) $3x + 5x - 2x$
- b) $7a^{2} - 3a + 2a^{2} + 5a$
- c) $4xy + 2x - xy + 3x$
- d) $10n - 5 - 3n + 8$
Patterns and the nth Term
When a pattern always grows by the same rule, one expression can give you any position $n$ at once. Some problems call it the general term $a_n$: it is the explicit rule for the pattern.
Look at the list $3, 5, 7, 9, \ldots$ It grows by $2$ each time, so $a_n$ starts with $2n$. Check $n = 1$: $2 \cdot 1 = 2$, but the first term is $3$, so we add $1$. The rule is $a_n = 2n + 1$.
What if the differences are not constant? For $2, 5, 10, 17, \ldots$ the differences are $3, 5, 7$: they themselves grow steadily. That is the signature of an $n^2$ term. Compare with the square numbers $1, 4, 9, 16, \ldots$
Observe the following series of figures made with dots and determine the algebraic expression of the general term $a_{n}$ (where $n$ is the position of the figure). How…
Observe the following series of figures made with dots and determine the algebraic expression of the general term $a_{n}$ (where $n$ is the position of the figure). How many dots will the figure in position $n=50$ have?
A digital artist is creating a pixel mosaic that grows following a specific pattern. In stage $n=1$ they use 3 pixels, in stage $n=2$ they use 8 pixels, in stage $n=3$…
A digital artist is creating a pixel mosaic that grows following a specific pattern. In stage $n=1$ they use 3 pixels, in stage $n=2$ they use 8 pixels, in stage $n=3$ they use 15 pixels, and so on.
Observe the visual pattern and complete the following tasks:
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Find the algebraic expression (general term) that determines the number of pixels needed for any stage $n$.
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If the digital canvas has a limit of $1000$ pixels, could the artist complete stage $n=30$? Justify your answer by calculating the numerical value.
Adding and Subtracting Polynomials
To add polynomials, combine like terms. To subtract, subtract every term of the second polynomial, which means changing all its signs.
To multiply a polynomial by a number, multiply every term by it. So $2(x^2 - 3x) = 2x^2 - 6x$.
Perimeter is a nice use of this. The perimeter of a rectangle is the sum of all four sides: $P = 2 \cdot \text{length} + 2 \cdot \text{width}$.
Given the polynomials $A(x) = 4x^3 - 6x^2 + 5x$ and $B(x) = -2x^3 + 8x^2 - 7$, perform the following operations and order the resulting polynomial from highest to lowest…
Given the polynomials $A(x) = 4x^3 - 6x^2 + 5x$ and $B(x) = -2x^3 + 8x^2 - 7$, perform the following operations and order the resulting polynomial from highest to lowest degree:
- a) $A(x) + B(x)$
- b) $A(x) - B(x)$
- c) $2 \cdot A(x) + 3 \cdot B(x)$
We are designing a new tablet whose screen is rectangular. The base measures $x$ inches and the height measures $5$ inches less than the base. Write an algebraic…
We are designing a new tablet whose screen is rectangular. The base measures $x$ inches and the height measures $5$ inches less than the base. Write an algebraic expression for the perimeter of the screen and simplify it.
Multiplying Monomials and Polynomials
To multiply a monomial by a polynomial, use the distributive property: $a(b + c) = ab + ac$. To multiply two polynomials, multiply every term of the first by every term of the second, then combine like terms.
Distributing is also how sales tax works. If a ticket costs $t$ dollars with a service fee of $f$ dollars, and the tax is $8\%$ of the total, the total cost is $1.08(t + f)$, which equals $1.08t + 1.08f$. For $t = 12$ and $f = 3$ the total is $1.08 \cdot 15 = \$16.20$.
Reduce the following algebraic expression as much as possible by operating with monomials and removing the parentheses:…
Reduce the following algebraic expression as much as possible by operating with monomials and removing the parentheses:
$3x \cdot (2x^2) - 4x^2 \cdot (x - 5) + \frac{12x^4}{3x}$
A surveillance drone flies over a rectangular lot whose dimensions are given by the polynomials $L(x) = x^2 - 3x + 5$ for the length and $A(x) = 2x + 1$ for the width.…
A surveillance drone flies over a rectangular lot whose dimensions are given by the polynomials $L(x) = x^2 - 3x + 5$ for the length and $A(x) = 2x + 1$ for the width. Obtain the polynomial that represents the total area of the lot.
An electric scooter rental company charges a base fare of $\$B$ for unlocking and $\$0,15$ for each minute of use ($m$). Furthermore, if the user travels more than $5$…
An electric scooter rental company charges a base fare of $\$B$ for unlocking and $\$0,15$ for each minute of use ($m$). Furthermore, if the user travels more than $5$ miles, a safety surcharge of $\$S$ is applied. A $21\%$ sales tax must be applied to the sum of these amounts.
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Write the algebraic expression for the total cost $C$ of a trip if the user exceeds $5$ miles in distance.
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Simplify the previous expression by carrying out the possible operations.
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If the base fare is $\$1$ and the safety surcharge is $\$2$, how much will a person pay who uses the scooter for $20$ minutes and travels $6$ miles?
Expressions for Area
Area formulas turn shapes into expressions. A rectangle has area $\text{length} \times \text{width}$, and a circle of radius $r$ has area $\pi r^2$. For a combined shape, add the areas of the parts, or subtract the part that is missing.
A fraction of a length works the same way. If a basketball court is $y$ feet long and half as wide, its width is $\dfrac{y}{2}$ and its area is $y \cdot \dfrac{y}{2} = \dfrac{y^2}{2}$, or $\tfrac{1}{2}y^2$.
A community sports center has a rectangular basketball court. If the length of the court is $y$ feet and its width is half the length, write the polynomial that…
A community sports center has a rectangular basketball court. If the length of the court is $y$ feet and its width is half the length, write the polynomial that represents the area of the court as a function of $y$.
Calculate the algebraic expression that represents the area of the following geometric figure as a function of $x$. Simplify the result by combining like terms.
Calculate the algebraic expression that represents the area of the following geometric figure as a function of $x$. Simplify the result by combining like terms.
The new model of a smartphone has a rectangular glass screen with dimensions $x$ wide and $x+4$ long. Around the screen there is a protective frame or "bezel" that has a…
The new model of a smartphone has a rectangular glass screen with dimensions $x$ wide and $x+4$ long. Around the screen there is a protective frame or "bezel" that has a uniform thickness of $2$ in. on all its sides.
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Find the simplified polynomial that expresses the total area of the front of the phone (screen plus frame).
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Find the simplified polynomial that expresses only the area of the frame (the gray area).
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If the width of the screen $x$ is $6$ in., what is the area of the frame?
Special Products
Some products show up so often that they are worth memorizing. Some problems call them algebraic identities. They are the special products.
The picture shows a square of side $a + b$ cut into four pieces: an $a^2$ square, a $b^2$ square, and two $ab$ rectangles.
Special products are great for algebraic proofs. The method is: name the numbers with a variable, write the expression, expand, simplify, and then check with a real example.
Use algebraic identities to expand the following expressions without performing the multiplication step by step: a) $(3x + 5)^2$ b) $(2a - 4b)^2$ c)…
Use algebraic identities to expand the following expressions without performing the multiplication step by step:
- a) $(3x + 5)^2$
- b) $(2a - 4b)^2$
- c) $(x^2 + 3) \cdot (x^2 - 3)$
Operate and simplify the following algebraic expression as much as possible using algebraic identities to reduce terms: $$(3x - 5)^2 - (3x + 5)(3x - 5) - 10(3 - 3x)$$…
Operate and simplify the following algebraic expression as much as possible using algebraic identities to reduce terms:
$$(3x - 5)^2 - (3x + 5)(3x - 5) - 10(3 - 3x)$$Show the step-by-step process, indicating which algebraic identity you have applied in each case.
Prove algebraically that the difference between the squares of two consecutive integers is always equal to the sum of these numbers. Instructions:* 1. Call the first…
Prove algebraically that the difference between the squares of two consecutive integers is always equal to the sum of these numbers.
Instructions:
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Call the first number $n$ and the next one $n+1$.
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Write the expression for the difference of their squares: $(n+1)^2 - n^2$.
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Expand and simplify the expression.
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Check the result with the numbers $12$ and $13$.
Factoring Out the GCF
Factoring is the reverse of distributing: you write a sum as a product. The simplest kind is pulling out the greatest common factor (GCF).
Factoring also works with the special products in reverse:
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$a^2 - b^2 = (a + b)(a - b)$, for example $x^2 - 49 = (x + 7)(x - 7)$.
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$a^2 + 2ab + b^2 = (a + b)^2$, for example $x^2 + 10x + 25 = (x + 5)^2$.
Letters like $\pi$ are just numbers, so you can factor them too. A square inscribed in a circle of radius $r$ has area $2r^2$. The area inside the circle but outside the square is $\pi r^2 - 2r^2 = r^2(\pi - 2)$.
Simplifying fractions with variables
A fraction with variables is a rational expression (some problems call it an algebraic fraction). Factor the top and the bottom, then cancel common factors.
Factor out the greatest common factor in the following algebraic expressions: a) $15x^4 - 10x^3 + 5x^2$ b) $6a^2b - 9ab^2 + 3ab$ c) $12x^2y^3 - 18x^3y^2$
Factor out the greatest common factor in the following algebraic expressions:
- a) $15x^4 - 10x^3 + 5x^2$
- b) $6a^2b - 9ab^2 + 3ab$
- c) $12x^2y^3 - 18x^3y^2$
A play area in a park is to be designed consisting of a large circle of radius $R$. Inside this circle, a square sandbox will be installed whose side measures exactly…
A play area in a park is to be designed consisting of a large circle of radius $R$. Inside this circle, a square sandbox will be installed whose side measures exactly the radius $R$ of the circle. The rest of the circle will be covered with artificial grass.
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Write the simplified algebraic expression for the area of the artificial grass zone as a function of $R$. Use $\pi$ as a symbol.
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Factor out the common factor in the expression obtained.
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If the radius $R$ is $10$ feet, calculate the grass area (use $\pi \approx 3,14$).
Simplify the following algebraic fraction by first factoring out the common factor and using algebraic identities in the numerator and the denominator: $$\frac{2x^3 -…
Simplify the following algebraic fraction by first factoring out the common factor and using algebraic identities in the numerator and the denominator:
$$\frac{2x^3 - 8x}{x^2 + 4x + 4}$$For what value of $x$ would this expression not make sense? (Hint: the denominator cannot be zero).
Back to the start: If the streaming plan costs \$15 a month ($x = 15$), the value of $\dfrac{x}{3}$ is $\dfrac{15}{3} = 5$, so each person pays $\$5$.