Adding and multiplying expressions

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 $50$ 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 $P(x) = 2x^2 - ax + 5$ 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:

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.

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

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 $-4$.

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 $-b$.

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

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

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

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) $2x(x+1) - 3(x+1)$

Aoife 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 Aoife 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 Aoife's by substituting the value $x = 2$ into the initial expression $(3x - 5)^2$, into Aoife's erroneous result, and into your simplified final polynomial. What conclusion do you draw?

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.

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:

• Aoife 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.”

• Seán 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: $(2n + 3)^2 - (2n + 1)^2$.”

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

1. a) Explain why Aoife's inductive method does not constitute a valid mathematical proof for all infinitely many odd numbers.
2. b) Work through Seán'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 Saoirse 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.