Polynomials

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

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?

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

• $5xy^2$

• $-x^2y$

• $2x^2$

• $\frac{1}{2}x^2y$

• $5x^2y^2$

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.

In a technology store, the price of a mid-range smartphone is $x$ pesos. A high-end smartphone costs double the mid-range one plus 9 000 pesos, and a tablet costs 4 800 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 $x = 12\,000$ pesos, how much will the total order cost?

A pizza delivery driver earns a fixed wage of $900$ pesos for each evening of work, plus an extra $180$ pesos for each home delivery order. The expression that calculates their earnings is $G = 900 + 180x$, 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.

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 $x = -1$.

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 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.

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 $C_2(x)$.

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

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

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

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

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

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

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

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.

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:

$(x + 2)^2$

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) $(\frac{1}{2}x + 6) \times (\frac{1}{2}x - 6)$

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

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 $x - 1$.

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.

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) $\frac{1}{3}x^4 - \frac{2}{3}x^2$

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 $x$.

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) $S(x) = 2x^2 + 20x + 50$