Sum and Product of Polynomials

Complete the following table, identifying the components of each monomial:

Given the polynomials $P(x) = 4x^2 - 3x + 5$ and $Q(x) = x^2 + 7x - 2$, perform the following operations by grouping like terms:

1. a) $P(x) + Q(x)$
2. b) $P(x) - Q(x)$

Find the polynomial expressing the perimeter of the following geometric figure (an L-block from a video game). Remember that the perimeter is the sum of all its sides.

A streamer charges a fixed amount of $3\,000$ pesos per livestream and, in addition, receives $120$ pesos for each new subscription ($s$) and $30$ pesos for each donation ($d$).

1. Write the algebraic expression (polynomial) representing their total income in one day.

2. If in a livestream they get $20$ subscriptions and $40$ donations, how much money have they earned in total?

Given the polynomials $P(x) = 2x^2 - ax + 5$ and $Q(x) = bx^2 + 3x - c$, determine the values of the parameters $a$, $b$ and $c$ knowing that adding them yields the identity polynomial $S(x) = 5x^2 + x + 2$. Justify your answer by comparing the coefficients of the terms of the same degree.

Perform the following multiplications of monomials, applying index laws:

1. a) $(3x^2) \times (4x^3) =$
2. b) $(-2a^2b) \times (5ab^3) =$
3. c) $(\frac{1}{2}x) \times (6x^5) =$

Apply the distributive property to calculate the area of the following rectangle, expressing the result as a polynomial:

Simplify the following polynomial expressions combining addition, subtraction and simple multiplication:

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

In a clothing store, the price of a hoodie is given by the polynomial $P(x) = 120x^2 + 600$, where $x$ is a logistics variable. If we buy $3x$ hoodies:

1. What is the polynomial that expresses the total cost of the purchase?

2. If for a value of $x=2$, the price of one hoodie is in pesos, how much will the $3x$ hoodies cost for that same value?

Given the polynomials $P(x) = 3x^2 - 5x + 2$ and $Q(x) = -x^2 + 2x - 7$, perform the following combined operations and simplify the resulting polynomial:

1. a) $2 \times P(x) + 3 \times Q(x)$
2. b) $P(x) - [Q(x) + (x^2 - 1)]$
3. c) $3x \times Q(x) + 5x^2$

Apply the distributive property to multiply the following polynomials and collect like terms:

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

A company that manufactures tablets has designed a new model whose screen has the dimensions shown in the following figure. The outer border (bezel) has a constant width of $2$ cm on all sides.

1. a) Express the total area of the tablet (including the bezel) using a polynomial.
2. b) Determine the polynomial representing the area of the screen alone (shaded area).
3. c) Calculate the polynomial representing the area of the tablet's bezel.

In the "AppZ" app store, the monthly revenue from selling a premium subscription is given by the polynomial $I(x) = 10x^2 + 50x$, where $x$ is the number of months since launch. On the other hand, maintenance costs are expressed as $G(x) = 2x^2 - 10x + 100$.

1. a) Find the polynomial $B(x)$ that represents the company's net profits ($Beneficio = Ingresos - Gastos$).
2. b) If the company decides to invest an extra amount in advertising equal to the product of $(x + 5) \times (x - 2)$, which is an additional cost, what would be the new total profit polynomial?

Find the perimetre of the following geometric figure in terms of the variable $x$, expressing the result as a simplified polynomial.

At a music festival, total revenue depends on the number of season tickets sold, represented by the variable $x$. The following polynomial functions have been determined:

• Ticket revenue: $I(x) = 3x^2 + 50x - 100$

• Logistics and security expenses: $G(x) = 2x^2 - 15x + 500$

• Marketing expenses: $M(x) = 10x + 1200$

If the net profit of the festival is calculated by subtracting all expenses from the revenue, find the simplified polynomial $B(x)$ that represents the profit. Once obtained, calculate the actual profit if $x = 100$ season tickets are sold.

An architecture firm designs a rectangular town square containing a central circular fountain and a landscaped garden area. The total area of the square is defined by the product of its base $b(x) = 3x + 5$ and its height $h(x) = 2x - 1$. If the area of the fountain is given by the polynomial $F(x) = x^2 + 2x + 1$, find the polynomial that represents the area of the garden area (the remaining space of the square).

Work out the following combined operation and simplify the result as far as possible, ordering the polynomial in descending order of degree:

Pay special attention to expanding the algebraic identity and the negative sign affecting the square bracket.

A courier company uses boxes whose dimensions (in cm) are in arithmetic progression and depend on a variable $x$:

• Width: $x - 2$

• Length: $x + 1$

• Height: $x + 4$

Find the polynomial $V(x)$ expressing the volume of the box. Then, find the polynomial $S(x)$ representing the total surface area of the six faces of the box. Simplify both expressions.

Find the quadratic polynomial $P(x)$ such that it satisfies the following three conditions simultaneously:

1. Its leading coefficient is $3$.

2. It has no constant term.

3. When multiplied by the binomial $(x - 2)$, the coefficient of the degree $2$ term of the resulting product is equal to $-4$.

Consider any two polynomials, $A(x)$ of degree $n$ and $B(x)$ of degree $m$. If we know that the degree of the product $A(x) \times B(x)$ is $7$ and that the degree of the expression $(A(x) + B(x))^2$ is $8$, what are the individual degrees of $A(x)$ and $B(x)$? Reason your answer based on the properties of the degree of the sum and product of polynomials.

Factorise the following algebraic expressions by taking out the common factor to write them as the product of a monomial and a polynomial:

1. a) $15x^4 - 10x^3 + 5x^2$
2. b) $3a^2b - 6ab^2 + 9ab$
3. c) $2x(x+1) - 3(x+1)$

Calculate the resulting polynomial of the following expression using standard algebraic identities (square of a sum, square of a difference, and product of a sum and a difference):

$A(x) = (2x + 3)^2 - (2x - 3)^2 + (x + 4) \times (x - 4)$

Simplify the final result as far as possible.

Determine the value of the coefficient $k$ so that the following equality between polynomials holds true:

$(x + 3) \times (2x - 5) = 2x^2 + kx - 15$

Once $k$ is found, work out the product $(x + 3) \times (2x - 5) \times (x + 1)$ and write the resulting polynomial in descending order of degree.

Expand and simplify the following expression involving three different variables ($x$, $y$, $z$). These types of expressions are common in physics formulae and data models:

Represent the final result as a simplified polynomial. What do you notice regarding the cross terms (such as $xy$ or $xz$)?