Operations with Polynomials

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?

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 $x = 1$

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 $10$.

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: $P(x) + Q(x)$.

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 $x$.

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: $C(x) - (2x^2 - x)$

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: $I(x) - G(x)$.

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 $C$?

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 $x = -2$.

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

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

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

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.

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

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

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: $x$

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).

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.

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) $(\frac{1}{2}x + 6)^2$

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 and simplify the following expression:

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

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

• a) $4x^2 + 8x$

• b) $5a^3 - 10a^2 + 15a$

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) $7x^5 - 14x^2$

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.

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:

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

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.