Exercises: Monomials and operations with monomials

In the world of algebra, not all expressions are "monomials". Remember that a monomial is a product of a number and one (or more) letters with natural number exponents. State which of the following expressions are monomials and which are not. If not, explain why:

1. a) $5x^3$
2. b) $2 + a$
3. c) $-3ab^2$
4. d) $\frac{4}{x}$
5. e) $x$
6. f) $\sqrt{x}$

Copy the following table into your notebook and complete the missing elements for each monomial. Be careful with signs and invisible exponents!

Two monomials are "like terms" if they have exactly the same algebraic part. In the list below, there are three hidden pairs of like monomials. Find and group them!

• $4x^2$

• $-5ab$

• $\frac{1}{3}x$

• $2x^2$

• $8ab$

• $-7x$

Simplify these expressions by adding or subtracting like monomials. If an expression cannot be simplified, explain why:

1. a) $3x + 5x - 2x$
2. b) $7a^2 - a^2$
3. c) $4x + 3y$
4. d) $5ab + ab - 3ab$
5. e) $x^3 + x^3 + x^3$

In an adventure video game, your character collects energy gems. Red gems are worth $x$ points each. If during the first level you get $5x$ gems, in the second level you find another $8x$, and in the third level you lose $3x$ due to a spell, simplify the expression to find out how many gems you have in total at the end.

To organise a party, you buy multipacks of soft drinks. Each pack contains $6$ cans and the price of one can is $y$ euros. If you buy $4$ identical packs, write the monomial that represents the total cost of the purchase.

A farmer has a rectangular vegetable patch to plant tomatoes. Help him calculate the expression for the area of the vegetable patch by multiplying the following monomials:

Perform the following multiplications of monomials, remembering to multiply the coefficients together and add the exponents of identical letters:

• a) $3 \times (4x)$

• b) $2x \times 5x$

• c) $(-3a^2) \times 2a$

• d) $4x \times 3y$

Aoife is reviewing operations with monomials and has written down these two statements in her notebook:

1) $4x^3 + 5x^3 = 9x^6$

2) $2a + 3b = 5ab$

Identify the conceptual error Aoife has made in each of the two operations. Explain, using the distributive property and the definition of like terms, why her reasoning is incorrect. Finally, write the corrected answers for both cases.

We want to discover the identity of an unknown monomial, which we will call $M$, about which we know the following clues:

1) Its coefficient is a negative integer strictly greater than $-5$.

2) Its literal part consists of the letters $x$ and $y$, both with an exponent greater than or equal to $1$.

3) Multiplying the unknown monomial $M$ by the monomial $3x^2y$ yields a monomial with coefficient $-12$ and a total degree of $6$.

Deduce and reasonably justify the coefficient and the possible values of the exponents of $M$. Write at least one monomial that satisfies all the conditions exactly, and then propose a different monomial that is similar to it.

In a strategy video game, resources are stored using algebraic expressions. Complete the following table so the programmer can organise the inventory correctly:

You are organising the files on your computer. To save space, you can only add or subtract "files" (monomials) that are like terms. Simplify the following expression as much as possible:

$7x^2 - 3x + 2x^2 - 10x + 5$

In the comic book store, $x$ represents an amount of manga. If Cian buys three times the square of that amount and then gets a discount of twice the square of that amount, what monomial represents the final amount of manga he paid for? Write down the operation and simplify.

A streaming server divides a data packet of size $24x^6$ among $4x^2$ users. Write the monomial that represents the amount of data each user receives.

We want to surround a shelf with a special polygonal shape with an LED strip. Calculate the monomial that represents the total perimeter of the figure by adding its sides:

Simplify the following divisions of monomials:

1. a) $\frac{15x^6}{3x^2} =$

1. b) $\frac{-20a^4}{5a^3} =$

1. c) $\frac{8y^3}{8y^3} =$

Apply the distributive property to expand the brackets and collect like terms in the following expression:

$3 \times (2x + 4) + 5 \times (x - 2)$

Find the expression for the area of the following rectangle as a monomial. Remember that area is calculated by multiplying the base by the height ($A = b \times h$).

At a Met Éireann weather station in the Wicklow Mountains, snowfalls are analyzed to evaluate the weight supported by building roofs. In a storage warehouse with a flat rectangular roof, the length of the warehouse is $4x$ meters, the width is $3x$ meters, and the thickness of the snow layer accumulated after a storm is measured as $0.05x$ meters.

1. a) Calculate the simplified monomial that models the volume of snow deposited on the roof (in $\text{m}^3$). Clearly specify its coefficient, its literal part, and its degree.
2. b) Knowing that each cubic meter of this snow has an average mass of $120\text{ kg}$, obtain the simplified monomial that represents the total mass of snow accumulated on the roof.
3. c) If a technical inspection determines that $x = 2\text{ meters}$, calculate the actual total mass of snow supported by the warehouse and reflect on whether the calculated roof dimensions ($4x$ in length by $3x$ in width) are realistic.

We propose analyzing the following combined operation with monomials:

1. a) Estimate without calculating: Before carrying out the numerical operations step by step, reason what the degree of the resulting monomial from the first block (the algebraic fraction) must be and what sign its coefficient will have. From this, deduce whether it will be possible to subtract the term $- 5x^4$ from it.
2. b) Check by calculating: Solve the expression step by step respecting the order of operations: first multiply the monomials in the numerator, then divide by the denominator, and finally perform the subtraction of like terms.
3. c) Evaluate: Compare the final result with your predictions from section a). Did the degree and the sign match what you had previously reasoned? Justify why the final result is a monomial and not an irreducible two-term expression.

A street artist is designing a mural with intertwined geometric shapes. The central design consists of a large rectangle with base $5x^2$ and height $3y$. Inside this rectangle, the artist leaves an unpainted gap in the shape of a triangle whose base measures $2x^2$ and whose height is $y$. Write the simplified monomial representing the area that will finally be painted in colour.

Look at the following tower built with decorative cubes. Each cube has an edge measuring $a$ cm. It is known that the top face of each cube is painted gold, and the outer side faces (which are not touching another cube) are painted silver.

1. Write the monomial representing the total area painted gold in a tower with $n$ storeys.

2. Write the monomial representing the total area painted silver if the tower has exactly 4 storeys like the one in the picture.

Complete the following "Monomial Pyramid". In this pyramid, the value of each box is the product of the two monomials directly below it. You must find the missing monomials in the base so that the monomial at the top is correct.

Make up two monomials, $M_1$ and $M_2$, that satisfy all of these conditions at the same time:

1. Both must have $a^2b$ as their literal part.

2. Their sum must be a monomial whose coefficient is $-10$.

3. When dividing $M_1$ by $M_2$, the result must be the number $4$.

Write down the two monomials and check that they satisfy the conditions.

Look at the design of this play area for a technology park. The local council needs to calculate the total area to install artificial turf of different colours in each section.

Calculate the following expressions using operations with monomials:

1. The area of Zone A.

2. The area of Zone B (formed by two identical triangles).

3. The total area of the compound simplifying the sum of the resulting monomials.

Look at the following series of figures built with white and grey square tiles. Mathematicians call these "growth patterns".

1. If $n$ represents the figure number, write a monomial representing the number of grey tiles in figure $n$.

2. Write a binomial for the number of white tiles in figure $n$.

3. Add both expressions to obtain the expression for the total number of tiles. Which figure would have a total of 101 tiles?

A "Mystery Monomial" has lost some of its parts after passing through an operation machine. Your mission is to reconstruct the original equation by finding the missing monomial in each case (represented by the symbol $?$):

1. a) $? + 7a^2b - 3a^2b = 10a^2b$

1. b) $(? \times 5x^4) : 2x^2 = 15x^5$

1. c) $\frac{24y^{10}}{?} = 4y^5$

Simplify the following combined expression, paying close attention to the order of operations and the laws of indices:

Explain step by step how you simplified each term before performing the final addition and subtraction. What is the degree of the resulting polynomial before total simplification?

In a review session, the teacher proposes simplifying the following expression with monomials:

Two students suggest two different strategies:

• Seán's Method: First solve the power in the parentheses in the numerator, then multiply all the monomials in the numerator together and those in the denominator together to form a single fraction, and finally divide the coefficients and subtract the exponents.

• Aoife's Method: After expanding the power in the parentheses, directly simplify the numerical factors and the powers with the same base shared by the numerator and the denominator before calculating the complete products.

1. a) Develop and calculate the expression step by step following Seán's Method.
2. b) Develop and calculate the expression step by step following Aoife's Method.
3. c) Compare both procedures: Do you arrive at the same result? What are the advantages and disadvantages of each strategy regarding the size of the intermediate numbers and the risk of error? Justify which one you would choose if the coefficients were much larger numbers.

A cardboard factory in Cork produces open-top boxes from a rectangular cardboard sheet whose dimensions are $6x$ meters long by $4x$ meters wide. To assemble each box, a square with side length $y$ meters (with $y < 2x$) is cut from each of the four corners of the sheet, and the four resulting flaps are folded upward.

1. a) Write the algebraic expression that represents the usable surface area of cardboard used to make the box (the initial area of the sheet minus the four cut-out corners). Justify whether this expression is a monomial or not according to the formal definition.
2. b) The design team establishes that, to always maintain optimal proportions, the corner cut must be exactly $y = 0.5x$:

1. Substitute this condition into the expression for the usable surface area and simplify it to obtain a single monomial. State its coefficient, literal part, and degree.

2. Determine the monomials that represent the base length, base width, and height of the box once folded.

3. Find the simplified monomial that models the inner volume of the box.

1. c) A customer orders a batch of boxes that can hold a minimum volume of $7500\text{ liters}$ ($7.5\text{ m}^3$). If the factory sets the machinery so that $x = 1\text{ meter}$, verify by calculating the volume using the obtained monomial whether the box meets the customer's order, and justify your answer.