Algebraic expressions
A music 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.
Monomials
Two monomials 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) $x + \frac{1}{2}x + 3x$
Polynomials
A polynomial is a sum of monomials that are not like terms (its terms). 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.
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).
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 root of the polynomial. On a graph, the roots 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 roots: $-1$, $1$ 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 $C(x+3)$.
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.