Monomials

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:

In a math app, a bot has incorrectly classified some monomials. Your task is to correct the following table by identifying the coefficient, the literal part, and the degree of each one. If you find any error in the bot's classification, write it correctly:

Monomial	Coefficient	Literal part	Degree
$-5x^2y$	$5$	$x^2y$	$2$
$\frac{3}{4}ab^3c$	$3$	$ab^3c$	$4$
$-x^4$	$-1$	$x$	$4$
$7$	$7$	None	$1$

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 $a = 3$, $b = -1$

It is known that the numerical value of the monomial $M(x, y) = -5x^a y^b$ is $-320$ when $x=2$ and $y=2$. Furthermore, it is known that the relative degree with respect to the variable $x$ is twice the relative degree with respect to $y$.

Find the exponents $a$ and $b$, justifying the steps followed.

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.

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

Simplify the following expression involving monomials as far as possible, applying the laws of indices and the order of operations:

$$\frac{(-3x^2y^3)^2 \times (2x^4y)^3}{(-6x^3y^2)^2}$$

Express the result as a single monomial.

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

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

Find the degree of this "chained" monomial and simplify it to its simplest form:

$$P(x) = x \times (2x^2)^3 : \left( \frac{1}{4}x^4 \right) + (-5x^3) \times (-2x^0)$$

Are the terms resulting from the sum like terms? Justify your answer.

Look at the following regular hexagon whose side measures $x$ meters. 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 center:

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 $d$.

Find the monomial representing the area of the following shaded region, which is part of the design of a new skatepark. Express the result in its simplest form.

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.

Calculate the monomial that represents the perimeter and the polynomial that represents the shaded area of the following composite figure (a large square with side length $3x$ from which four identical small squares with side length $y$ have been cut out at its corners):

Simplify the expressions as much as possible.

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?

In the following "product pyramid", each brick is the result of multiplying the two monomials directly below it. Complete the missing spaces by finding the monomials $A$, $B$ and $C$.

Three friends rent a padel court. The total cost is given by the algebraic expression $C(t, l) = 360t + 120l$, where $t$ is the time in hours and $l$ is the number of rented rackets.

1. a) If they decide to split the cost equally, write the algebraic expression representing what each person has to pay.

1. b) If a fourth friend joins the group, the total cost of the rackets increases (there are now $l+1$) and the expense is divided by four. Write the new expression for the individual payment.

1. c) If the playing time is $1.5$ hours and each person rents their own racket, how many pesos does each of the three original friends save when the fourth player joins?