Exercises: Operations with polynomials

Find the 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 $B(x) = -x^2 + 4x - 6$, find the resulting expression from performing the operation $2 \cdot A(x) - B(x)$.

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

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

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

Expand the following expressions by applying the corresponding 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: $(x + 3)(x - 3) - (x - 2)^2$.

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

A cardboard box without a lid is to be made from a rectangular sheet $30 \text{ cm}$ long by $20 \text{ cm}$ wide. To do this, four identical squares of side $x \text{ cm}$ are cut out from each of the corners and the flaps are folded up. Find the polynomial $V(x)$ that expresses the volume of the resulting box in terms of $x$.

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

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

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

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 the profit is equal to total revenue (quantity times price) minus total costs. What will the profit be if $40$ units are manufactured and sold?

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

The height in feet of a drone as a function of time in seconds is given by the polynomial $H(t) = -5t^2 + 20t + 2$. A second drone follows a path described by $G(t) = -5t^2 + 15t + 10$.

Find the polynomial $D(t)$ that represents the difference in height between the first and second drone ($D(t) = H(t) - G(t)$). At what time $t$ are both drones at the same height?

Determine if the polynomial $P(x) = x^4 - 5x^3 + 7x^2 - 5x + 6$ is divisible by the binomial $(x^2 + 1)$ by performing the division. If so, write the factorization of $P(x)$ as a product of two polynomials.

Find the values of the parameters $a$ and $b$ so 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 running track consists of a central rectangular region of length $3x$ and width $2x$, with two semicircles at its ends of diameter $2x$. Express the total area and the outer perimeter of the track as polynomials in terms of $x$ (leave $\pi$ 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 triangle (that is, they satisfy the Pythagorean theorem). Check your result by evaluating the polynomials for $x = 4$.

A startup estimates that its monthly revenue follows the polynomial $I(x) = -2x^3 + 60x^2$ and its operating costs are $C(x) = 15x^2 + 400x$, where $x$ is the number of months elapsed since its creation.

Find the profit polynomial $B(x)$. Given 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 is zero again.

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

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

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

Determine the values of the coefficients $m$ and $n$ so 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.

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

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