Exercises: Polynomials

In the world of video games, the scoring system of a new platform title works as follows: for each gold coin collected you receive an unknown amount of points $x$, but for each enemy defeated you receive three times as many points as for a coin. At the end of the level, you always get a fixed bonus of $50$ experience points.

Write the algebraic expression representing the total score of a player who has collected $12$ coins and defeated $4$ enemies.

Copy and complete the following table in your exercise book to identify the fundamental elements of these monomials:

To add or subtract monomials, they must be like terms. Select from the following list which ones are like terms to the monomial $5x^2y$:

• $5xy^2$

• $-x^2y$

• $2x^2$

• $\frac{1}{2}x^2y$

• $5x^2y^2$

A pizza delivery driver earns a fixed wage of $300$ pesos for each evening of work, plus an extra $60$ pesos for each home delivery order. The expression that calculates their earnings is $G = 300 + 60x$, where $x$ is the number of orders.

Calculate the numerical value of their earnings (the total amount they will receive) if one evening they deliver $8$ orders.

Analyse the following polynomial and answer the questions:

$P(x) = -2x^3 + 5x^2 - 8$

1. a) What is the degree of the polynomial?
2. b) What is the name of the term that has no variable ($-8$)?
3. c) How many terms does it have? Based on the number of terms, is it a monomial, binomial or trinomial?
4. d) What is the coefficient of the highest degree term?

Find the simplified algebraic expression representing the perimeter of the following rectangular photo frame:

Given the following polynomials, perform the addition $A(x) + B(x)$ and collect like terms:

$A(x) = 4x^2 - 5x + 3$

$B(x) = x^2 + 2x - 7$

Algebraic identities allow us to calculate certain powers quickly. Use the formula for the square of a sum $(a + b)^2 = a^2 + 2ab + b^2$ to expand the following expression:

$(x + 2)^2$

In an electronics store, the price of a mid-range smartphone is $x$ pesos. A high-end smartphone costs twice the mid-range one plus 3 000 pesos, and a tablet costs 1 600 pesos less than the mid-range smartphone.

1. a) Write the algebraic expression representing the total cost of buying two mid-range smartphones, one high-end smartphone, and three tablets.
2. b) Simplify the resulting expression to get a polynomial in its simplest form.
3. c) If the price of the basic smartphone is $x = 4\,000$ pesos, how much will the total order cost?

Given the following polynomial:

$P(x) = -5x^4 + \frac{2}{3}x^2 - x + 9$

State with reasoning the following elements:

1. a) The degree of the polynomial.
2. b) The coefficient of the degree 2 term.
3. c) The constant term.
4. d) The numerical value of $P(x)$ for $x = -1$.

A small water rocket is launched from a platform. The height $h$ (in meters) of the rocket as a function of time $t$ (in seconds) is given by the polynomial:

$h(t) = -5t^2 + 25t + 2$

1. a) At what height is the launch platform? (Hint: calculate $h(0)$).
2. b) At what height will the rocket be 2 seconds after being launched?
3. c) Will the rocket be higher at 3 seconds or at 4 seconds? Justify your answer by calculating the corresponding numerical values.

Perform the following operations with polynomials and simplify the final result:

1. a) $(4x^3 - 5x^2 + x - 7) + (x^3 + 2x^2 - 10)$
2. b) $(3x^2 - 6x + 2) - (x^2 - 4x - 5)$
3. c) $2 \times (x^3 - 3x + 1) + 4 \times (x^2 - 2)$

Calculate the area of the shaded region in the following figure. Express the result as a simplified polynomial in terms of $x$.

Multiply the following polynomials using the distributive property and combine like terms:

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

Apply the algebraic identity formulas to expand the following expressions:

1. a) $(3x + 4)^2$
2. b) $(2x^2 - 5)^2$
3. c) $(x + 7) \times (x - 7)$
4. d) $(\frac{1}{2}x + 6) \times (\frac{1}{2}x - 6)$

Factorising out a common factor is the reverse process of the distributive property. Use this technique to simplify the following algebraic expressions:

1. a) $12x^5 - 8x^3 + 4x^2$
2. b) $5a^2b - 15ab^2 + 10ab$
3. c) $\frac{1}{3}x^4 - \frac{2}{3}x^2$

A video game designer is creating a tile pattern for a dungeon floor. On level 1 they use a central tile, and on each subsequent level they surround the previous pattern with a new layer of tiles of a different color. The total number of tiles on level $n$ follows a logical sequence.

Look at the figure showing the first three levels and:

1. Find the polynomial $P(n)$ that expresses the total number of tiles needed to build up to level $n$.

2. Calculate how many tiles a level $n = 12$ floor will have.

A courier company charges for sending packages according to the following rates: a fixed handling fee of $G$ pesos, plus a price per kilogram of weight $x$, and a "premium distance" surcharge equivalent to the square of the package's weight divided by 4.

1. Write the polynomial $C(x)$ representing the total cost of sending a package of $x$ kilograms.

2. If the handling fee is $G = 5$ pesos and we want to send a package of $8$ kg, what will the total cost be?

3. If the company decides to double the cost per kilogram and keep all other parameters the same, write the new polynomial $C_2(x)$.

Calculate the area of the shaded region in the following figure, expressing the result as a simplified polynomial in terms of $x$. The outer shape is a square with side $3x + 2$ and the inner cutout is a rectangle with base $x + 4$ and height $x - 1$.

Expand and fully simplify the following algebraic expression using algebraic identities and factorisation:

State whether the final result is a constant number or depends on the value of $x$.

Given the polynomial $P(x) = x^3 - kx^2 + 5x - 6$, determine the value of the coefficient $k$ knowing that the numerical value of the polynomial for $x = 2$ is equal to zero ($x = 2$ is a root of the polynomial). Once $k$ is found, calculate the numerical value of $P(x)$ for $x = -1$.

A park is rectangular in shape. Originally, its base measures $x$ meters and its height measures $5$ meters less than the base. The municipality decides to extend the park by increasing the base by $3$ meters and the height by $2$ meters.

1. Write the polynomial $A_1(x)$ for the area of the original park.

2. Write the polynomial $A_2(x)$ for the area of the extended park.

3. Find the polynomial representing the increase in area $I(x) = A_2(x) - A_1(x)$ and simplify it.

Perform the following combined operation with polynomials and collect like terms:

Take out the common factor and use the notable identities to factor the following polynomials (factorization).

Remember the notable identities: $a^2 + 2ab + b^2 = (a+b)^2$, $a^2 - 2ab + b^2 = (a-b)^2$ and $a^2 - b^2 = (a+b)(a-b)$.

1. a) $Q(x) = 3x^3 - 12x$

1. b) $R(x) = x^4 - 10x^3 + 25x^2$

1. c) $S(x) = 2x^2 + 20x + 50$