Operations with Polynomials

Calculate the numerical value of the polynomial $P(x) = -2x^3 + 5x^2 - 3x + 4$ for $x = -2$ and for $x = 1$.

Given the polynomials $A(x) = 3x^2 - 5x + 1$ and $B(x) = -x^2 + 4x - 6$, calculate the expression resulting from carrying out the operation $2 \times A(x) - B(x)$.

Multiply the polynomials $P(x) = x^2 - 3x + 2$ and $Q(x) = 2x - 5$. Express the result ordered from highest to lowest degree.

Expand the following expressions using the appropriate algebraic identities:

1. a) $(3x + 4)^2$
2. b) $(2x - 5)^2$

Express the following binomials as a product of two factors using the difference of two squares identity:

1. a) $16x^2 - 25$
2. b) $9x^4 - 49$

Expand and fully simplify the following algebraic expression: $(x + 3)(x - 3) - (x - 2)^2$.

Expand and fully simplify the following algebraic expression, paying special attention to signs and algebraic identities:

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

Divide the polynomial $D(x) = 4x^4 - 2x^3 + 6x^2 - x + 5$ by the binomial $d(x) = x^2 + 2$. Clearly state the quotient polynomial $C(x)$ and the remainder polynomial $R(x)$.

Check whether the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by carrying out the corresponding division. If so, write the factorisation of $P(x)$ as the product of two polynomials.

Determine the values of the coefficients $m$ and $n$ such that the division of the polynomial $D(x) = 4x^4 - 2x^3 + mx^2 + nx - 5$ by the divisor $d(x) = 2x^2 - x + 3$ is exact.

Use Ruffini's rule to divide the polynomial $P(x) = 3x^3 - 5x^2 + 2x - 7$ by $(x - 2)$. Write down the quotient and remainder obtained.

Knowing that dividing a polynomial $P(x)$ by $(x - 3)$ gives a quotient of $C(x) = 4x^2 - 2x + 1$ and a remainder of $R = -7$, reconstruct the polynomial $P(x)$ and express it in expanded form.

Calculate the value of the parameter $k$ such that the division of the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$ by the binomial $(x + 2)$ has a remainder equal to $10$. Use the remainder theorem to justify your answer.

Find the values of the parameters $a$ and $b$ such that the polynomial $P(x) = x^3 + ax^2 + bx - 6$ is exactly divisible by $(x - 2)$ and, at the same time, leaves a remainder of $-12$ when divided by $(x + 1)$.

A polynomial $P(x)$ of degree 3 has roots $x = 1$, $x = -2$ and $x = 4$. Given also that the value of the polynomial for $x = 0$ is equal to $16$, find the expanded expression of $P(x)$ (that is, in the form $ax^3 + bx^2 + cx + d$).

A rectangular garden has dimensions of $(2x + 5)$ meters long and $(2x - 5)$ meters wide. Inside it, a square fountain with side length $(x - 1)$ meters will be built, and the rest will be covered with lawn. Express the area of the lawn area as a simplified polynomial in $x$.

An open-topped cardboard box is to be made from a rectangular sheet measuring $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares with side length $x \text{ cm}$ are cut out from each corner and the flaps are folded upwards. Find the polynomial $V(x)$ that expresses the volume of the resulting box in terms of $x$.

The production cost of $x$ units of an electronic device is given by the polynomial $C(x) = 50x + 200$. The retail price of each unit depends on the quantity manufactured and follows the expression $p(x) = 300 - 2x$.

Find the profit polynomial $B(x)$, knowing that profit is equal to total revenue (quantity times price) minus total costs. What will the profit be if $40$ units are manufactured and sold?

The height in meters of a ball thrown vertically as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second ball, thrown from a different point, follows a trajectory described by $G(t) = -5t^2 + 15t + 10$.

Find the polynomial $D(t)$ representing the difference in height between the first and second balls ($D(t) = H(t) - G(t)$). At what instant $t$ are both balls at the same height?

A running track consists of a central rectangular region of length $3x$ and width $2x$, capped at its ends by two semicircles of diameter $2x$. Express the total area and outer perimeter of the track as polynomials in terms of $x$ (leave the number $\pi$ indicated in the coefficients).

Prove algebraically that, for any real number $x > 1$, the expressions $a = x^2 - 1$, $b = 2x$ and $c = x^2 + 1$ always form the sides of a right-angled triangle (that is, they satisfy Pythagoras' theorem). Check your result by evaluating the polynomials for $x = 4$.

A startup estimates that its monthly revenue follows the polynomial $I(x) = -2x^3 + 75x^2$ and its operating costs are $C(x) = 15x^2 + 400x$, where $x$ is the number of months elapsed since its founding.

Find the profit polynomial $B(x)$. Knowing that in month $x = 10$ the profit is exactly zero (break-even point), factor the polynomial $B(x)$ to determine in which other month the profit becomes zero again.

Simplify the following algebraic expression, factorising the final result if possible:

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

Next, evaluate the resulting polynomial for $x = 2$ and $y = -1$.

A water tank is cylindrical with a base radius of $(x + 2)$ meters and a height of $(2x - 1)$ meters. A solid cubic block with edge length $x$ meters is submerged inside it. Express, using an ordered polynomial, the volume of water needed to fill the remaining space in the tank. Leave the constant $\pi$ indicated in any coefficients that require it.