Terms

In an app store, the profit per game download is expressed by the monomial $7x^2y$. Identify the following elements of this expression:

• Coefficient:

• Literal part:

• Degree of the monomial:

We are organising an inventory of robotics parts. Group the following monomials into pairs of like terms:

$5x^2$, $-3ab$, $x$, $\frac{1}{2}ab$, $-2x^2$, $8x$

Calculate the numerical value of the monomial $M(x) = -5x^2$ for the indicated values of the variable:

1. a) $x = 2$
2. b) $x = -1$
3. c) $x = 0$

Work out the numerical value of the following monomials for the values of the variables given in each case:

1. a) $P(x, y) = -3x^2y$ for $x = -2$, $y = 5$

1. b) $Q(a, b) = \frac{2}{3}a^3b^2$ for $a = 3$, $b = -1$

Given the monomial $P(a, b) = -4a^2b^3$, we want to calculate its numerical value for $a = -2$ and $b = -1$.

1. a) Estimate the sign: Before performing any full numerical calculation, determine with reasoning whether the final result will be positive or negative. To do this, analyze separately the sign of the coefficient $-4$, of the power $(-2)^2$, and of the power $(-1)^3$.
2. b) Estimate the order of magnitude: Without calculating the exact value, deduce whether the absolute value of the result will be greater or less than $100$. Justify your mental estimation.
3. c) Check: Substitute the values into the monomial, calculate the final result step by step respecting the order of operations, and confirm whether your predictions from the previous sections were correct.

Simplify the following algebraic expressions by adding and subtracting like monomials:

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

A content creator is analysing the growth of their social media. The number of "Likes" on their photos is given by the monomial $8x^2y$, and the number of "Shares" by $3x^2y$. During the last month, they received an extra number of views equivalent to $\frac{1}{2}x^2y$.

Calculate the simplified algebraic expression representing the total interaction by adding the three items.

Aoife has to simplify the expression $5x^3 + 3x^3$ in a class exercise and writes on the board: $5x^3 + 3x^3 = 8x^6$. When asked by the teacher, she justifies her answer saying: "I added the coefficients ($5 + 3 = 8$) and I also added the exponents of the letters ($3 + 3 = 6$)".

1. a) Identify the mistake Aoife made and explain, using the distributive property (factoring out the literal part), why the exponents should not be added.
2. b) Write the correct result of the addition.
3. c) What operation between the monomials $5x^3$ and $3x^3$ would have had to be performed for the result to be a monomial of degree $6$? Justify your answer.

To calculate the numerical value of the algebraic expression:

$$E = 7x^2y - 12x^2y + 8x^2y - 2x^2y$$

for $x = -\frac{2}{3}$ and $y = 9$, two students follow different paths:

• Method A (Cian): Directly substitutes the values of $x$ and $y$ into each of the four monomials separately, calculates the four numerical results, and then performs the additions and subtractions with the numbers obtained.

• Method B (Aoife): First combines like terms to reduce the entire expression to a single monomial, and then substitutes the numerical values only once.

1. a) Solve the exercise in detail following Cian's method and then following Aoife's.
2. b) Analyze both procedures: in which one did you have to do more operations with powers and fractions? Justify which of the two methods is more convenient and safer to avoid calculation errors.

Perform the following monomial multiplications and simplify the result:

1. a) $3 \times (5x^4)$
2. b) $(-2x^2) \times (4x)$
3. c) $(3a) \times (2b)$

In a strategy video game, your base's energy increases according to the following monomial operation. Simplify the expression to a single monomial:

$$(12x^5 \times 3x^2) : (9x^4)$$

What is the degree of the resulting monomial?

Simplify the following expression combining powers and divisions of monomials:

$$\frac{(-2a^3b)^2 \times 5ab^4}{10a^4b^3}$$

Fill in the gaps with the required monomial to make the equalities correct:

1. a) $7x^3y + \dots \dots = 10x^3y$

1. b) $(-2ab^2) \times (\dots \dots) = 12a^3b^2$

1. c) $\frac{15x^4y^2}{\dots \dots} = 3x^2y$

Look at the following figure formed by two adjacent rectangles.

1. a) Write the monomial that represents the area of each rectangle separately.
2. b) Write the polynomial that represents the total area.
3. c) Factor out the common factor in the expression for the total area.

Factorise the following algebraic expression by taking out the highest common factor of the terms. Take special care with fractional coefficients:

$$\frac{4}{9}a^5b^2 - \frac{8}{27}a^3b^4 + \frac{2}{3}a^3b^2$$

Observe the following sequence of patterns made of small squares. The first term ($n=1$) has 4 squares, the second ($n=2$) has 7, and the third ($n=3$) has 10.

1. a) Determine the expression that represents the number of squares in terms of position $n$, knowing that the relationship is linear (a monomial in $n$ plus a constant term).

1. b) How many squares will the pattern in position $n=150$ have?

1. c) If a pattern has 91 squares, what position $n$ does it occupy in the sequence?

Look at the following regular hexagon whose side measures $x$ metres. Write the monomial representing its perimeter and calculate its value if each side were $4.5$ cm.

Find the monomial expressing the area of the following rectangle intended for a gaming area in a shopping centre:

Translate the following phrases into algebraic language using a single monomial:

1. a) Double the cube of a number $x$.
2. b) Half the product of two numbers $a$ and $b$.
3. c) Three times the square of the distance $d$.