Exercises: Operations with polynomials

In a strategy video game, your inventory is divided into two chests. The contents of the first chest are given by the polynomial $P(x) = 5x^2 + 3x - 4$ and the second by $Q(x) = 2x^2 - 5x + 8$, where $x$ represents the rarity level of the items. Calculate the total polynomial that represents the sum of the two chests: $P(x) + Q(x)$.

To organize an e-sports event, a company uses the expression $G(x) = 10x^2 - 4x + 100$ to calculate its expenses and $I(x) = 15x^2 + 2x$ to calculate its revenue (where $x$ is the number of sponsors). Find the polynomial that represents the net profit, knowing that profit is the difference between revenue and expenses: $I(x) - G(x)$.

Calculate the numerical value of the polynomial $A(x) = 3x^3 - 5x^2 + 2x - 7$ for the following cases:

• a) For $x = 0$

• b) For $x = 1$

Look at the following diagram of a rectangular skate park. Write the simplified polynomial that represents the perimeter of the skate park in terms of $x$.

Carry out the following operation by multiplying the monomial by the polynomial (applying the distributive property):

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

Identify the main elements of the following polynomial:

$P(x) = -7x^4 + 2x^3 - x + 12$

• a) What is its degree?

• b) What is the leading coefficient?

• c) What is the constant term?

• d) How many terms does it have?

A young designer is creating a logo for their clothing brand. The central logo is a rectangle like the one in the figure. Find the polynomial that represents the total area of the figure.

Factor out the common factor in the following simple algebraic expressions:

• a) $4x^2 + 8x$

• b) $5a^3 - 10a^2 + 15a$

In a video game store, the inventory of three different categories is expressed by the following polynomials, where $x$ represents the number of new releases this month:

• Action: $A(x) = 3x^2 + 5x - 8$

• Adventure: $B(x) = -x^2 + 2x + 10$

• Strategy: $C(x) = 2x^2 - 4x + 1$

Calculate the following expressions, simplifying the result:

1. a) The total number of video games in the store: $A(x) + B(x) + C(x)$
2. b) The difference between Action and Adventure games: $A(x) - B(x)$
3. c) How many Strategy games would be left if half of a batch of size $4x^2 - 2x$ is sold? Calculate: $C(x) - (2x^2 - x)$

Calculate the area of the shaded region in the following figure. The figure represents a rectangular metal plate from which an inner rectangle has been removed to leave space for an electronic component. Express the result as a polynomial in terms of $x$.

Additional data:

• Outer base: $5x + 3$

• Outer height: $3x$

• Inner base: $x + 2$

• Inner height: $x$

Carry out the following polynomial multiplications and combine like terms:

1. a) $3x \cdot (2x^2 - 5x + 4)$
2. b) $(-2x^2) \cdot (x^3 - 3x^2 + 7)$
3. c) $(x + 3) \cdot (x^2 - 2x + 5)$
4. d) $(2x - 1) \cdot (3x^2 + x - 2)$

Given the following combined expression with polynomials, calculate and simplify to obtain a single polynomial written in descending order of degree:

$P(x) = 2x \cdot (x^2 - 3x + 1) - (x - 4) \cdot (2x + 5)$

An architect designs a square plaza of side $L = 2x + 5$ feet. In the center, they decide to place a fountain, also square, of side $l = x - 3$ feet.

1. a) Expand the expression for the area of the plaza using algebraic identities: $(2x + 5)^2$
2. b) Expand the expression for the area of the fountain using algebraic identities: $(x - 3)^2$
3. c) Calculate the polynomial that represents the surface area of the plaza left free for walking (Plaza area - Fountain area).

Apply algebraic identities to expand the following expressions directly, without carrying out the multiplication step by step:

1. a) $(3x + 4)^2$
2. b) $(5x - 2y)^2$
3. c) $(2x^2 + 1) \cdot (2x^2 - 1)$
4. d) $(\frac{1}{2}x + 6)^2$

Factoring out the common factor is the reverse process of the distributive property. Analyze the following algebraic expressions and factor out all possible common factors (numbers and letters):

1. a) $12x^4 - 8x^3 + 4x^2$
2. b) $15a^3b^2 - 10a^2b$
3. c) $9x^2y - 3xy^2 + 6xy$
4. d) $7x^5 - 14x^2$

In a laboratory, the growth of a bacterial population $A$ follows the polynomial $A(t) = 2t^2 + 10t + 50$, where $t$ is the time in hours. Another population $B$ grows according to the polynomial $B(t) = t^2 - 5t$.

If the scientists need to know the behavior of a mixed culture defined by the operation $C(t) = 3 \cdot A(t) - 2 \cdot B(t)$, what is the simplified polynomial that represents this population $C$?

Calculate the following expression, fully simplifying the resulting polynomial. Pay special attention to the order of operations and the use of parentheses with negative signs in front:

$(2x^2 - 5) \cdot (x + 3) - (x - 4)^2 + 3x \cdot (x^2 - 2x + 1)$

In the center of a square garden of side $2x + 5$ feet, a rectangular swimming pool has been installed whose dimensions are $x + 2$ feet long and $x - 1$ feet wide. Find the polynomial that represents the area of the lawn region (the part of the garden not occupied by the pool).

Determine the unknown polynomial $Q(x)$ in the following equation:

$(-3x^3 + 5x^2 - 7) + Q(x) = x^4 - 2x^2 + 5x - 1$

Next, calculate the numerical value of $Q(x)$ for $x = -2$.

A shipping company designs boxes in the shape of a rectangular prism. The dimensions of the box are as follows: the length measures $x + 5$ in, the width is $x - 2$ in, and the height is twice the width.

1. a) Express the volume of the box as a polynomial in terms of $x$.
2. b) Expand and simplify this polynomial.

Find the value of the coefficient $k$ in the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$, given that its numerical value for $x = 2$ is $10$.

Apply algebraic identities to expand and simplify the following expression:

$3 \cdot (2x + \frac{1}{2})^2 - 2 \cdot (3x - 1) \cdot (3x + 1)$

Simplify the following algebraic fraction by first factoring out the common factor in the numerator and the denominator:

$\frac{12x^4 - 18x^3 + 6x^2}{6x^3 - 6x^2}$

Hint: Remember that a polynomial can be divided by a monomial by dividing each of its terms.

A cell phone case manufacturer has determined that the cost of producing $x$ units is given by the polynomial $C(x) = 0,5x^2 + 10x + 200$, while the revenue obtained from selling those same units is expressed as $I(x) = -0,2x^2 + 50x$.

1. a) Find the polynomial $B(x)$ that represents the profit (Revenue minus Costs).
2. b) Work out the profit if $40$ cases are sold.