Algebraic expressions
A music streaming subscription costs €9 a month. How much do you pay in $x$ months? $9x$ euros. With one letter you have written, all at once, the answer for any number of months. That is algebra: using letters to represent numbers that can change or that we do not know yet.
Algebraic language
An algebraic expression combines numbers and letters (variables) using operations. Translating from everyday language into algebraic language is the first skill you need:
| Everyday language | Algebraic language |
|---|---|
| Twice a number | $2x$ |
| A number plus 5 | $x + 5$ |
| The square of a number | $x^2$ |
| Half of a number, minus 3 | $\frac{x}{2} - 3$ |
| Half of (a number minus 3) | $\frac{x - 3}{2}$ |
| Two consecutive numbers | $x,\ x + 1$ |
| An even / odd number | $2n$ / $2n + 1$ |
In a mobile app store, the cost of a monthly subscription to a music service is $x$ euros. In addition, there is a fixed fee of $1.50$ € for management expenses on each…
In a mobile app store, the cost of a monthly subscription to a music service is $x$ euros. In addition, there is a fixed fee of $1.50$ € for management expenses on each bill (one bill is issued per person per month). Express in algebraic language:
- a) The total cost of one month's subscription.
- b) The cost of three months' subscription for the same user.
- c) The cost of one month's subscription for a group of 4 friends if each of them receives a discount of $2$ € on the base price $x$.
An architect is designing a rectangular room. The length of the room is $5$ metres more than the width. If we let $x$ be the width of the room: a) Write an algebraic…
An architect is designing a rectangular room. The length of the room is $5$ metres more than the width. If we let $x$ be the width of the room:
- a) Write an algebraic expression representing the perimeter of the room.
- b) Write an algebraic expression representing the area of the room.
Terms and coefficients
Two terms are like terms if they have the same literal part: $3x^2$ and $-7x^2$ are; $3x^2$ and $3x$ are not. Only like terms can be added or subtracted, by adding their coefficients:
$$3x^2 - 7x^2 + x^2 = (3 - 7 + 1)x^2 = -3x^2$$Copy and complete the following table, identifying the components of each given monomial: | Monomial | Coefficient | Variable Part | Degree | | :--- | :---: | :---: |…
Copy and complete the following table, identifying the components of each given monomial:
| Monomial | Coefficient | Variable Part | Degree |
|---|---|---|---|
| $-5x^2y$ | |||
| $\frac{3}{4}ab^3$ | |||
| $x^2yz^2$ | |||
| $-12$ |
Simplify the following algebraic expressions by adding or subtracting like terms: a) $3x^2 - 5x + 2x^2 + x - 4x^2$ b) $2ab^2 + 3a^2b - 5ab^2 + a^2b$ c)…
Simplify the following algebraic expressions by adding or subtracting like terms:
- a) $3x^2 - 5x + 2x^2 + x - 4x^2$
- b) $2ab^2 + 3a^2b - 5ab^2 + a^2b$
- c) $x + \frac{1}{2}x + 3x$
Linear, quadratic and cubic expressions
An expression made of several terms added or subtracted is called a polynomial. It may contain like terms before you simplify it; for example, $3x^2 - 5x + 4$ is a quadratic polynomial. If the highest degree is 1, 2 or 3 we call it a linear, quadratic or cubic expression. It is normally written in order from highest to lowest degree:
$$P(x) = -2x^3 + 5x^2 - x + 7$$-
Degree of the polynomial: the highest degree of its terms (here, 3).
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Leading coefficient: the coefficient of the term of highest degree ($-2$).
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Constant term: the term without $x$ ($7$).
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If some degree is missing, the polynomial is incomplete: $x^3 - 4$ has no $x^2$ or $x$ term.
Squares and the difference of two squares
Sometimes you multiply brackets by the same pattern again and again. Take $(x + 3)^2$. It means $(x + 3)(x + 3)$, so a square of side $x + 3$ splits into four parts: a square $x^2$, two rectangles $3x$ and a square $9$.
Reading the patterns backwards factorises: $x^2 + 10x + 25 = (x + 5)^2$.
Using the notable identities (square of a sum, square of a difference and sum times difference), expand or factor the following algebraic expressions as appropriate.…
Using the notable identities (square of a sum, square of a difference and sum times difference), expand or factor the following algebraic expressions as appropriate. Relate part 2 to the geometric interpretation shown in the image.
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Expand: $(3x + 2)^2$
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Expand: $(x - 4)^2$
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Expand: $(2x + 5)(2x - 5)$
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Factor the polynomial (write it as the square of a binomial): $x^2 + 10x + 25$
Multiplying expressions
To multiply two expressions, multiply every term of the first by every term of the second (the distributive law), then collect like terms.
Consider the following polynomials representing the dimensions and properties of different geometric solids: $$ P(x) = 3x^2 - 5x + 1 $$ $$ Q(x) = 2x^2 + 4x - 2 $$ $$…
Consider the following polynomials representing the dimensions and properties of different geometric solids:
$$ P(x) = 3x^2 - 5x + 1 $$ $$ Q(x) = 2x^2 + 4x - 2 $$ $$ R(x) = x - 3 $$Perform the following combined operations and simplify the final result:
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Calculate the difference: $P(x) - Q(x)$.
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Calculate the product: $Q(x) \times R(x)$.
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Calculate: $2 \times P(x) - 3 \times Q(x)$.
Taking out a common factor
To factorise, take out the highest common factor (HCF) of all the terms. Each term of $x^3 + 4x^2 + 3x$ contains $x$, so the expression equals $x(x^2 + 4x + 3)$. Multiplying out gives the original again.
Factorising $ax^2 + bx + c$ (Higher Level only)
When the coefficient of $x^2$ is not 1, find two numbers with sum $b$ and product $a \times c$. Use them to split the middle term, then factorise by grouping.
The area of a rectangle is given by the polynomial $A(x) = 2x^2 + 7x + 3$. 1. Factorise the polynomial $A(x)$ to find the algebraic expressions corresponding to the base…
The area of a rectangle is given by the polynomial $A(x) = 2x^2 + 7x + 3$.
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Factorise the polynomial $A(x)$ to find the algebraic expressions corresponding to the base and height of the rectangle, knowing that both expressions are binomials with integer coefficients.
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If we know that the value of $x$ is such that $x > 0$, calculate what the perimeter of the rectangle would be in terms of $x$. Simplify the resulting expression as much as possible.
We have a cardboard box in the shape of a cuboid (rectangular prism) whose volume is given by the algebraic expression: $$ V(x) = x^3 + 4x^2 + 3x $$ 1. Factorise out the…
We have a cardboard box in the shape of a cuboid (rectangular prism) whose volume is given by the algebraic expression:
$$ V(x) = x^3 + 4x^2 + 3x $$-
Factorise out the common factor in the expression for the volume.
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Factorise completely the resulting polynomial to find the algebraic expressions corresponding to the length, width and height of the box.
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If we know that $x = 5$ cm, what are the actual numerical dimensions of the box?
Given the polynomial $P(x) = -2x^3 + 5x^2 - x + 7$, answer the following questions: a) What is the degree of the polynomial? b) What is the coefficient of the term of…
Given the polynomial $P(x) = -2x^3 + 5x^2 - x + 7$, answer the following questions:
- a) What is the degree of the polynomial?
- b) What is the coefficient of the term of degree 2?
- c) What is the constant term?
- d) Order the polynomial in ascending order (from lowest to highest degree).
Perform the following additions and subtractions of polynomials: a) $(3x^2 - 5x + 4) + (2x^2 + x - 6)$ b) $(x^3 - 2x^2 + 3) - (x^2 - 5x + 1)$ c)…
Perform the following additions and subtractions of polynomials:
- a) $(3x^2 - 5x + 4) + (2x^2 + x - 6)$
- b) $(x^3 - 2x^2 + 3) - (x^2 - 5x + 1)$
- c) $(-4x^4 + 2x^2) + (4x^4 - x^2 + x)$
Design and write down in each case an algebraic expression that strictly satisfies all the required conditions: a) A monomial involving two different variables, $x$ and…
Design and write down in each case an algebraic expression that strictly satisfies all the required conditions:
- a) A monomial involving two different variables, $x$ and $y$, whose coefficient is a negative fraction and whose total degree is equal to $5$.
- b) A polynomial of degree $3$ with the indeterminate $x$, ordered from highest to lowest degree, that has three terms, whose leading coefficient is $4$, its constant term is $-6$, and has $x = 1$ as a root (remember that a number $a$ is a root of a polynomial if its numerical value is $0$, that is, $P(a) = 0$). Justify step by step how you deduce the missing middle term.
Numerical value and roots
The numerical value of a polynomial for $x = a$ is what you get by substituting $x$ with $a$ and calculating. It is written $P(a)$.
If $P(a) = 0$, we say that $a$ is a solution of the equation $P(x) = 0$. On a graph, the solutions are the points where the graph crosses the $x$-axis. Run your finger along the graph to see the numerical value at each point:
Here $P(x) = x^3 - 2x^2 - x + 2$ has three solutions of $P(x) = 0$: $-1$, $1$ and $2$. Check it: $P(1) = 1 - 2 - 1 + 2 = 0$.
Calculate the numerical value of the following algebraic expressions for the given values of the variables: a) $3x^2 - 5x + 2$ for $x = 2$ b) $2(a + b)^2$ for $a = -1$…
Calculate the numerical value of the following algebraic expressions for the given values of the variables:
- a) $3x^2 - 5x + 2$ for $x = 2$
- b) $2(a + b)^2$ for $a = -1$ and $b = 3$
- c) $\frac{x + 2y}{2}$ for $x = 4$ and $y = -3$
An electric scooter rental company uses the following algebraic expression to calculate the rental cost in cents based on the minutes of use ($x$): $$ C(x) = 2x^2 - 5x +…
An electric scooter rental company uses the following algebraic expression to calculate the rental cost in cents based on the minutes of use ($x$):
$$ C(x) = 2x^2 - 5x + 150 $$Note: This formula is a promotional model for VIP users and is valid for short journeys.
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Calculate the numerical value of the cost if you use the scooter for 10 minutes.
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If a user has paid exactly 150 cents, how many minutes in theory has he been using the scooter according to the formula? (Hint: look at the constant term and find all possible solutions).
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Expand the cost expression for a time of $(x + 3)$ minutes, that is, calculate and simplify $C(x+3)$.
At a Met Éireann mountain weather station, the temperature $T$ (in degrees Celsius) during a winter day is modeled using the following polynomial: $$T(h) = -0.2h^2 +…
At a Met Éireann mountain weather station, the temperature $T$ (in degrees Celsius) during a winter day is modeled using the following polynomial:
$$T(h) = -0.2h^2 + 2.4h - 3$$where $h$ represents the number of hours elapsed since 08:00 in the morning ($h = 0$ corresponds to 08:00, $h = 4$ corresponds to 12:00, etc.).
- a) Observe the coefficients of the polynomial before calculating. Knowing that at 08:00 ($h = 0$) the temperature was $-3$ ºC, make a reasoned estimate of whether at 12:00 ($h = 4$) the temperature will have exceeded $0$ ºC or not.
- b) Now calculate the exact numerical value $T(4)$ and compare the result with your previous estimate.
- c) A novice observer claims that at $18:00$ ($h = 10$) the temperature will reach $15$ ºC. Without calculating the exact value yet, estimate whether this claim is plausible taking into account the sign of the quadratic term $-0.2h^2$ for increasing values of $h$. Then, calculate $T(10)$ and confirm whether the claim was correct.
We want to check whether certain values are "roots" of a polynomial. Remember that a root is a value that makes the polynomial equal to zero. Given the polynomial…
We want to check whether certain values are "roots" of a polynomial. Remember that a root is a value that makes the polynomial equal to zero.
Given the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$:
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Use Ruffini's rule to divide $P(x)$ by $(x - 1)$. What is the remainder? Is $x=1$ a root of the polynomial?
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Without using Ruffini, calculate the numerical value $P(-2)$. Is $x=-2$ a root?
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Write the polynomial $P(x)$ as the product of its irreducible factors given its roots.