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 that of the second by $Q(x) = 2x^2 - 5x + 8$, where $x$ represents the rarity level of the items. Calculate the total polynomial representing the sum of the two chests: $P(x) + Q(x)$.

To organize an esports 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 income (where $x$ is the number of sponsors). Find the polynomial that represents the net profit, knowing that profit is the difference between income and expenses: $I(x) - G(x)$.

Evaluate 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 track. Write the simplified polynomial that represents the perimetre of the track in terms of $x$.

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

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

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 his 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.

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

• a) $4x^2 + 8x$

• b) $5a^3 - 10a^2 + 15a$

Aoife solved the following polynomial subtraction in her notebook: $(5x^2 - 3x + 4) - (2x^2 - 7x - 1)$. In the final step, she wrote $3x^2 - 10x + 3$ as the result. Explain which two sign errors she made when applying the rule for subtracting polynomials, detail why they occur, and write out the correct step-by-step expansion until reaching the final simplified polynomial.

You are asked to calculate the product of the polynomials $P(x) = 3x^2 - 2$ and $Q(x) = 2x^2 + 5x - 4$. Answer the following questions before and after performing the full multiplication:

1. a) Before multiplying term by term, deduce what the degree of the resulting polynomial $P(x) \times Q(x)$ must be and what its constant term must be. Explain your reasoning.
2. b) Your classmate Aoife claims without checking the calculations that the result is $6x^5 + 15x^3 - 4x^2 - 10x - 8$. Without expanding the product, what two immediate reasons based on your previous deduction prove that her result is impossible?
3. c) Calculate the full product by applying the distributive property, combine like terms, and check whether your estimation from section a) was correct.

In a video game shop, 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 shop: $A(x) + B(x) + C(x)$
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 room 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: $x$

Perform the following polynomial multiplications and collect like terms:

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

Given the following combined polynomial expression, expand and simplify to obtain a single polynomial arranged in descending order of degree:

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

An architect designs a square town square with a side length of $L = 2x + 5$ metres. In the centre, they decide to place a square fountain, also with a side length of $l = x - 3$ metres.

1. a) Expand the expression for the area of the square 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 representing the area of the square that remains free for walking (Area of square - Area of fountain).

Apply algebraic identities to expand the following expressions directly, without multiplying step by step:

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

Factorising out the common factor is the reverse process of the distributive property. Analyse the following algebraic expressions and factorise 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 scientists need to know the behaviour of a mixed culture defined by the operation $C(t) = 3 \times A(t) - 2 \times B(t)$, what is the simplified polynomial that represents this population $C$?

When erasing the board at the end of maths class, the polynomial $P(x)$ that completed the following equality vanished: $(2x^2 - 3x + 5) - P(x) + 3x \times (x - 2) = -x^2 + x + 9$.

1. a) Without guessing random numbers, isolate $P(x)$ algebraically on one side of the equality.
2. b) Expand the products and combine like terms to obtain the polynomial $P(x)$ in its simplified standard form.
3. c) Check that your answer is correct by substituting the polynomial $P(x)$ you obtained into the left-hand side of the original equality and verifying that it matches the right-hand side.

The municipal electric bicycle service of an Irish city such as Cork calculates its monthly costs and revenue per bicycle as a function of $x$, where $x$ represents hundreds of kilometres travelled per month (for example, $x = 2$ corresponds to $200\text{ km}$):

• Battery and motor cost: $B(x) = 2x^2 + 15x + 30$ (in euros).

• Brakes and tyres cost: $F(x) = x^2 + 25x + 10$ (in euros).

• Revenue from heavy-use surcharges: $I(x) = 5x \times (x + 4)$ (in euros).

1. a) Model the total maintenance cost of a bicycle, $G(x)$, by adding $B(x)$ and $F(x)$ into a single simplified expression.
2. b) What real economic meaning does the constant term of $G(x)$ have if a bicycle spends the entire month parked at the station ($x = 0$)?
3. c) Model the net balance of the bicycle using the polynomial $N(x) = I(x) - G(x)$. Carry out the subtraction paying special attention to signs and simplify the result.
4. d) If a bicycle travels $300\text{ km}$ in a month ($x = 3$), calculate the numerical value $N(3)$ and interpret whether that month the bicycle generates a positive balance (profit) or a negative one (loss) for these items.

Work out the following expression, simplifying the resulting polynomial as much as possible. Pay special attention to the order of operations and the use of brackets with negative signs in front of them:

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

In the centre of a square garden with a side length of $2x + 5$ metres, a rectangular swimming pool has been installed with dimensions of $x + 2$ metres in length and $x - 1$ metres in width. Find the polynomial that represents the area of the lawn (the part of the garden not occupied by the pool).

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

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

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

A parcel company designs cuboid-shaped boxes. The dimensions of the box are as follows: the length is $x + 5$ cm, the width is $x - 2$ cm, and the height is double 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 $P(x) = 2x^3 - kx^2 + 5x - 4$, knowing that evaluating its value for $x = 2$ gives a result of $10$.

Apply algebraic identities to expand and simplify the following expression:

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

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

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

A manufacturer of phone cases 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).

1. b) Calculate the profit if $40$ cases are sold.

To calculate the numerical value of the algebraic expression $E(x) = (2x - 6) \times (x^2 + 5x - 2) - (2x - 6) \times (x^2 + 3x + 8)$ when $x = 103$, two students propose different approaches:

• Seán's method: Multiply the first two polynomials together, then multiply the second two, subtract the two resulting polynomials by changing all the signs of the second one, combine like terms, and substitute $x = 103$.

• Aoife's method: Notice that the binomial $(2x - 6)$ is repeated in both blocks, factor it out from the entire expression, simplify what remains inside the brackets, and only at the end substitute $x = 103$.

1. a) Apply Seán's method by fully expanding and simplifying the expression $E(x)$.
2. b) Apply Aoife's method by factoring out $(2x - 6)$, simplify the resulting expression, and check that you arrive at exactly the same simplified polynomial as in section a).
3. c) Justify reasonably which of the two methods is more convenient to evaluate at $x = 103$ while minimising the number of operations and the risk of error. Use that more advantageous method to find the value of $E(103)$ mentally.

Consider the product of polynomials $P(x) = (a \times x + 3) \times (2x^2 + b \times x - 4)$, where $a$ and $b$ are unknown integer constants. When expanding this product and grouping all its like terms, we know that the resulting polynomial satisfies two conditions:

1. Its leading coefficient (the number multiplying $x^3$) is equal to $6$.

2. The degree 2 term vanishes completely (its final coefficient is $0$).

1. a) Write the expression obtained from multiplying $(a \times x + 3) \times (2x^2 + b \times x - 4)$ term by term before grouping like terms.
2. b) Deduce reasonably the exact value of the numbers $a$ and $b$ from the two given conditions.
3. c) Using the values found, write the polynomial $P(x)$ fully expanded, simplified, and arranged in descending order of degree.
4. d) Factor out the greatest common monomial factor possible from the resulting polynomial $P(x)$ and explain how you would check that your factorisation is correct using the distributive property.