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 $15$ euros for each evening of work, plus an extra $3$ euros for each home delivery order. The expression that calculates their earnings is $G = 15 + 3x$, 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:

$(x + 2)^2$

Cian has to calculate the numerical value of the polynomial $P(x) = 3x^2 - 4x + 5$ for $x = -2$. In his notebook, he writes the following resolution:

When he finishes, Cian adds: "Furthermore, it is an incomplete polynomial because it has no term with $x^3$".

1. a) Find and explain in detail the two mistakes made by Cian (one arithmetic mistake in the calculation of powers or signs, and one conceptual mistake in the classification of the polynomial).
2. b) Carry out the correct calculation to find $P(-2)$ and classify the polynomial by stating its degree, whether it is complete or incomplete, its leading coefficient, and its constant term.

Design a polynomial $P(x)$ that simultaneously satisfies the following four conditions:

1. Its degree is $3$.

2. It is an incomplete polynomial and does not contain a degree $1$ term.

3. Its leading coefficient is $-2$.

4. Its constant term is $16$.

Once you have written your polynomial:

1. a) Check whether the number $x = 2$ is a root of your polynomial by calculating its numerical value $P(2)$ and justifying your conclusion based on the definition of a root.
2. b) Without performing lengthy numerical substitutions, reason why it is impossible for the number $x = 0$ to be a root of this polynomial, relying on the value of its constant term.

In a technology store, the price of a mid-range smartphone is $x$ euros. A high-end smartphone costs double the mid-range one plus 150 €, and a tablet costs 80 € 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 $x = 200$ €, 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 metres) 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 public electric bicycle rental company in Galway charges non-subscribers a monthly fee that depends on usage time. The amount is calculated using the polynomial $C(t) = 0.05t^2 + 1.2t + 8$, where $t$ represents the total number of hours of use per month and $C(t)$ is the resulting cost in euros.

1. a) Interpret what the constant term of the polynomial represents in real life by calculating the numerical value $C(0)$.
2. b) A user logs $10$ hours of use in the month of April and doubles their time to $20$ hours in the month of May. Calculate the bill amount for each of these two months.
3. c) Looking at the results for both months and the terms of the polynomial, algebraically justify why doubling the hours of use does not exactly double the total cost, indicating which term is responsible for this effect.

In a school technology project, a small water rocket is launched vertically. Its height in metres above the ground $t$ seconds after launch is given by the polynomial $h(t) = -5t^2 + 29t + 6$.

1. a) Without calculating the exact value, make an estimate by mentally rounding the coefficient $29$ to $30$ to predict whether at $t = 2$ seconds the rocket's height will be above or below $40$ metres. Explain in one sentence how you reasoned your estimate.
2. b) Now calculate the exact numerical value of $h(2)$ and compare it with your previous prediction. Do you consider that your estimate was appropriate and useful? Justify your answer.
3. c) When calculating $h(6)$, exactly $0$ is obtained. Interpret what this result physically means for the flight of the rocket and what mathematical name the number $6$ receives with respect to this polynomial.

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 colour. 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 shipping packages according to the following rate: a fixed handling fee of $G$ euros, plus a price of $2$ € for each kilogram of weight, and a "premium distance" surcharge equal to the square of the package's weight divided by 4.

1. Write the polynomial $C(x)$ that represents the total cost of shipping a package of $x$ kilograms.

2. If the handling fees are $G = 5$ € and we want to send an $8$ kg package, what will the total cost be?

3. If the company decides to double the cost per kilogram and keep the rest of the parameters unchanged, 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 $x$ metres and its height measures $5$ metres less than the base. The local council decides to extend the park by increasing the base by $3$ metres and the height by $2$ metres.

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

To check whether the numbers $x = 3$ and $x = -1$ are roots of the polynomial expressed in factored form $P(x) = (x - 3) \times (x^2 + 5x + 4)$, two students propose different approaches:

• Seán's strategy: Multiply all terms to obtain the polynomial in its expanded, ordered, and simplified form, and then substitute $x = 3$ and $x = -1$.

• Aoife's strategy: Directly substitute the values into each pair of parentheses separately without expanding the multiplication, checking whether any of the factors equals zero.

1. a) Apply Aoife's strategy to check whether $x = 3$ is a root of $P(x)$, showing the steps followed.
2. b) Apply whichever strategy you consider most convenient to check whether $x = -1$ is a root of $P(x)$.
3. c) Compare both approaches and argue which of the two methods is more efficient (fewer calculations and lower risk of sign errors). What mathematical property of multiplication does Aoife's strategy rely on to avoid calculating the second pair of parentheses when the first is equal to zero?

Consider the family of quadratic polynomials described by the expression $P(x) = x^2 + b \times x + 15$, where $b$ represents an unknown integer.

1. a) Knowing that the graph of this polynomial intersects the horizontal $X$-axis at the point with abscissa $x = -3$, deduce with reasoning the value that coefficient $b$ must have. (Recall the relationship between the intersection points with the $X$-axis and the roots of the polynomial).
2. b) With the value of $b$ you obtained, write the final expression of the polynomial and check whether it also intersects the $X$-axis at the point $x = -5$.
3. c) A classmate claims that if we change the value of $b$ to another suitable integer, we could make $x = 0$ a root of this polynomial. Prove algebraically that your classmate is wrong and explain what specific change would have to be made to the polynomial so that $x = 0$ could be a root.