Exercises: Addition and multiplication of polynomials

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

A streamer charges a fixed amount of $50$ euros per livestream and, in addition, receives $2$ euros for each new subscription ($s$) and $0,5$ euros 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 $P(x) = 4x^2 - 3x + 5$ 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.

Perform the following multiplications of monomials, applying index laws:

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

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

In a clothing shop, the price of a hoodie is given by the polynomial $P(x) = 2x^2 + 10$, 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 euros, how much will the $3x$ hoodies cost for that same value?

Lucía has to expand the expression $(3x - 5)^2$ on an exam and writes the following on her paper:

1. a) Identify and explain with reasoning the two conceptual errors that Lucía made when applying the notable identities.
2. b) Expand the expression correctly by applying the formula for the square of a difference step by step.
3. c) Numerically verify your result against Lucía's by substituting the value $x = 2$ into the initial expression $(3x - 5)^2$, into Lucía's erroneous result, and into your simplified final polynomial. What conclusion do you draw?

To simplify the expression $(2x + 7)\cdot(2x - 7)$, two classmates propose solving it in different ways:

• David proposes: “I multiply term by term using the distributive property: $2x \cdot 2x + 2x \cdot (-7) + 7 \cdot 2x + 7 \cdot (-7)$, and then I combine like terms.”

• Sara states: “Since it is a sum multiplied by a difference, I can directly apply the notable identity $(a + b)(a - b) = a^2 - b^2$, directly writing $(2x)^2 - 7^2$.”

1. a) Apply both procedures step by step and check whether both lead to the same resulting polynomial.
2. b) Justify which of the two strategies is more efficient and explain which terms always cancel out and why when we multiply the sum of two terms by their difference.

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

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.

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

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

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 $(x + 5) \cdot (x - 2)$, which is an additional cost, what would be the new total profit polynomial?

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):

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

Simplify the final result as far as possible.

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

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

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

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

In a municipal community garden project, a neighborhood association has a rectangular plot whose dimensions are $(2x + 3)$ meters long and $(x + 5)$ meters wide. To optimize space, the city council authorizes adding on one of the sides a covered rectangular area for composting and tools that measures $(x - 1)$ meters wide and shares the same length, $(2x + 3)$ meters.

1. a) Write the reduced polynomial $A_T(x)$ that represents the total surface area occupied by the garden and the new covered area together.
2. b) Determine the simplified algebraic expression of the polynomial $D(x)$ indicating by how many square meters the area of the original garden exceeds the area of the composting zone.
3. c) If the measurements from urban planning regulations correspond to $x = 4\text{ m}$, numerically calculate the total area of the facility and determine what percentage of this total area is represented by the composting zone.

Consider the following combined operation between polynomials: $P(x) = (3x^2 - 2)\cdot(2x^3 + 5x - 4) - (x^3 + 1)^2$.

1. a) Before carrying out any algebraic calculations on paper, answer these questions by justifying your prediction:

1. What must be the degree of the product $(3x^2 - 2)\cdot(2x^3 + 5x - 4)$ and the coefficient of its leading term?

2. What must be the degree of the expansion of $(x^3 + 1)^2$?

3. When subtracting both expressions, will the highest-degree monomials cancel out, or what do you predict the final degree of polynomial $P(x)$ to be?

4. What will be the constant term of $P(x)$ by analyzing only the numbers without $x$ from each part?

1. b) Now carry out the product, the expansion of the square of the sum, and the subtraction, combining like terms to obtain the reduced form of $P(x)$.
2. c) Compare your final polynomial with the four estimates you made in section a) and confirm whether your previous deductions were correct.

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.

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.

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).

Consider any two polynomials, $A(x)$ of degree $n$ and $B(x)$ of degree $m$. If we know that the degree of the product $A(x) \cdot B(x)$ 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.

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 $-4$.

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 $xz$)?

It is known that expanding and reducing the polynomial expression $A(x) = (ax + 3)^2 - (bx - 1)(bx + 1) + c$, where $a$, $b$, and $c$ are positive integers, gives the exact polynomial $R(x) = 7x^2 + 24x + 15$.

1. a) Algebraically expand the expression for $A(x)$ in terms of the letters $a$, $b$, and $c$, clearly grouping the resulting monomials by their degree ($x^2$, $x$, and constant term).
2. b) Set up the equalities that must hold between the coefficients of $A(x)$ and those of $R(x)$, and deduce with justification the exact values of $a$, $b$, and $c$.
3. c) If the condition that $b$ must be a positive number were removed and it were allowed to be any integer, would there be another valid value for $b$? Justify your answer by algebraically analyzing what happens to the product $(bx - 1)(bx + 1)$ if you substitute $b$ with its opposite $-b$.

We want to prove the following general mathematical property: “The difference between the squares of two consecutive odd numbers is always a multiple of $8$.” In class, three different methodological approaches are discussed:

• Alba says: “It is enough to test several examples: $3^2 - 1^2 = 8$, $5^2 - 3^2 = 16$, $7^2 - 5^2 = 24$. Since in all of them it gives a multiple of $8$, the property is proven.”

• Bruno proposes: “Let's write two general consecutive odd numbers as $(2n + 3)$ and $(2n + 1)$, where $n$ is any integer, and algebraically expand the difference of their squares: $(2n + 3)^2 - (2n + 1)^2$.”

• Clara suggests: “Instead of expanding each square, let's use the difference of two squares identity in reverse on Bruno's expression: $A^2 - B^2 = (A + B)(A - B)$, setting $A = 2n + 3$ and $B = 2n + 1$.”

1. a) Explain why Alba's inductive method does not constitute a valid mathematical proof for all infinitely many odd numbers.
2. b) Work through Bruno's method by applying the formulas for notable identities, combine like terms, and factor out the common factor to prove that the result is necessarily divisible by $8$.
3. c) Work through the method proposed by Clara and compare both algebraic paths: explain with reasoning which of the two requires fewer calculation steps and explain the advantages of transforming a difference of squares into a product.