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) $\sqrt{x}$

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) $x^3 + x^3 + x^3$

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:

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

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:

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

• a) $3 \cdot (4x)$

• b) $2x \cdot 5x$

• c) $(-3a^2) \cdot 2a$

• d) $4x \cdot 3y$

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:

Simplify the following divisions of monomials:

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

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

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

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.

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

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

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 ($A = b \cdot h$).

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.