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ESO Y4B · Mathematics
Polynomials · Polynomials

Algebraic expressions

A fitness app calculates the calories burned using a formula of the type $C(t) = 0{,}15t^2 + 8t$, 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 having to repeat the calculation each time. This block works with expressions of this kind, increasingly elaborate, and with the tools for operating on them: the starting point for everything that comes next in Bachillerato.

Before you start, make sure you know: Algebraic expressions (3rd ESO)
Before you start, make sure you know: Operations with powers (3rd ESO)

Monomials: coefficient, literal part and degree

$$M(x,y) = -\frac{2}{3}x^4y^2$$ Diagram of a monomial showing its coefficient, its literal part and the calculation of the degree by adding the exponents. −2/3 x⁴y² coefficient literal part degree = 4 + 2 = 6(sum of exponents)
Practice★☆☆
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: $M(x,y) = -\frac{2}{3}x^4y^2$.

Solve

Polynomials: degree and numerical value

Not every expression with letters is a polynomial: the exponents of the literal part must be natural numbers, and the letter cannot appear in a denominator or inside a root.

Diagram of a polynomial identifying its terms, coefficients, constant term and the degree of the polynomial. 3x² − 5x + 2 is a polynomial 5/x x^(1/2) are not: letter in denominator or non-natural exponent
Practice★☆☆
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 $x = -2$.

Solve
Practice★☆☆
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:

  • $A(x) = 3x^2 - \frac{5}{x}$

  • $B(x) = \sqrt{2}x^3 - x$

  • $C(x) = 4x^{1/2} + 1$

Solve

Operations with monomials

Only monomials that are like terms (same literal part) can be added or subtracted, by combining their coefficients; to multiply or divide monomials they do not need to be like terms: you work with the coefficients on one side and with each letter on the other, adding or subtracting their exponents.

Visual comparison between adding like monomials and multiplying monomials with coefficients and exponents. adding like terms 3x²y − 5x²y + 8x²y = 6x²y product and quotient (they do not need to be like terms) (−3a²b³)·(4ab⁴) = −12a³b⁷ coefficients are multiplied, exponents are added
Practice★☆☆
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: $3x^2y - 5x^2y + 8x^2y$.

Solve
Practice★☆☆
Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.

Multiply the following monomials: $(-3a^2b^3) \cdot (4ab^4)$.

Solve
Practice★★☆
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: $\frac{-24x^5y^4z}{6x^2y^3}$.

Solve

From everyday language to algebraic language

Translating a statement into an algebraic expression is done in stages: first each operation on a letter, then how they are combined.

Diagram of the translation of everyday phrases into algebraic operations and expressions. "the cube of a"minus"the square of b" a³−b² a³ − b²

Often the statement also gives a numerical value and asks you to find a parameter: the condition is translated into an equation and solved.

Practice★☆☆
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'.

Solve
Practice★☆☆
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 $5$.

Solve
Practice★★☆
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 $-4$.

Solve

Algebraic expressions in real-life problems

Perimeters, areas, rates or energies are naturally written as algebraic expressions when some measurement has not yet been fixed.

Geometric figure of a circle inscribed in a square, used to calculate the area between the two figures by subtraction. side = 2x radius = x shaded area = (2x)² − πx²
Practice★★☆
A clothes shop offers a discount of $20\%$ on all its items. If you buy $x$ shirts that originally cost $25$ euros each and $y$ pairs of trousers that cost $40$ euros…

A clothes shop offers a discount of $20\%$ on all its items. If you buy $x$ shirts that originally cost $25$ euros each and $y$ pairs of trousers that cost $40$ euros each, write the algebraic expression for the total cost of the purchase. Calculate the cost if you buy $3$ shirts and $2$ pairs of trousers.

Solve
Practice★★☆
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 metres 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 metres per second. Calculate the kinetic energy of a $800$ kg vehicle travelling at $20$ m/s.

Solve
Practice★★☆
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).

x 2x
Solve

Back to the start: If you exercise for $30$ minutes, the app calculates the calories burned by finding the numerical value for $t = 30$: $C(30) = 0{,}15 \cdot 30^2 + 8 \cdot 30 = 0{,}15 \cdot 900 + 240 = 135 + 240 = 375$ calories.

Summary

Let's practice! 10 medium exercises in this section
★★☆ A clothes shop offers a discount of $20\%$ on all its items. If you buy $x$ shirts that…★★☆ The kinetic energy of a moving body is given by the expression $E_c = \frac{1}{2}mv^2$,…★★☆ The figure shows a square with side length $2x$ containing inside it a circle of radius…★★☆ Calculate the numerical value of the algebraic expression…
Let's practice! 10 hard exercises in this section
★★★ An open-topped box is to be made from a rectangular cardboard sheet measuring $40$ cm…★★★ Prove algebraically that the difference between the squares of two consecutive odd…★★★ Given the algebraic expression $E(a,b) = \frac{a^3 - b^3}{a^2 + ab + b^2}$, simplify it…★★★ In an electrical circuit, the equivalent resistance $R_{eq}$ of three resistors $R_1$,…
See all the exercises in this section