Polynomial Division

Divide the following polynomial by the given monomial, simplifying each of the resulting terms:

The area of a rectangular recording studio is given by the polynomial $A(x) = 3x^2 + 15x$. If the width of the studio measures $3x$ meters, what is the algebraic expression that represents the length of the studio?

Use polynomial long division to find the quotient $C(x)$ and the remainder $R(x)$ of the following polynomial operation. Remember to order and complete the terms of the dividend before starting:

$(6x^4 + 7x^3 - 5x^2 + 1) : (3x^2 - x + 2)$

In any polynomial division, the relation $P(x) = d(x) \times c(x) + r(x)$ holds. If when dividing a polynomial by $d(x) = x + 3$ we have obtained a quotient of $c(x) = x - 1$ and a remainder of $r(x) = 5$, what is the dividend polynomial $P(x)$?

Reconstruct the dividend polynomial $D(x)$ knowing that when divided by $d(x) = x^2 - 1$, the quotient obtained is $C(x) = x + 5$ and the remainder is $R(x) = 3x - 2$. Use the fundamental property of division: $D(x) = d(x) \times C(x) + R(x)$.

When dividing a polynomial $P(x)$ by $Q(x) = x^2 - 4$, we obtain a quotient of $C(x) = x^2 + 2x - 3$ and a remainder of $R(x) = 5x + 1$. Reconstruct the original polynomial $P(x)$ and express it in standard form (ordered from highest to lowest degree).

A team of young programmers is designing a level for a video game where the volume of a supply crate is $V(x) = x^3 - 6x^2 + 12x - 8$. If the base of the crate has an area of $A(x) = x^2 - 4x + 4$, what is the expression for the height of the crate? What value does the height take if $x = 5$ units?

A rectangular billboard is to be designed with a total surface area defined by $S(x) = x^4 - 16$. If one of its sides measures $L(x) = x^2 + 4$, calculate the polynomial that represents the other side. Next, state whether the area is divisible by $(x - 2)$ and justify your answer without carrying out long division.

Apply Ruffini's rule to calculate the quotient and the remainder of the following division:

Use Ruffini's rule to divide the polynomial $P(x) = x^3 + 4x^2 - x - 4$ by the binomial $d(x) = x + 1$. Clearly state the quotient $C(x)$ and the remainder $R$.

Apply the Ruffini rule to calculate the quotient and the remainder of the division of the polynomial $P(x) = x^4 - 3x^2 + 5x - 10$ by the binomial $(x + 3)$.

Remember that, to apply the rule, the polynomial must be complete and ordered (if any term is missing, its coefficient is written as $0$) and that the value placed on the left is the root of the divisor.

1. a) Write the coefficients of the dividend and the value of $a$ that you use in the Ruffini rule.

1. b) Carry out the Ruffini rule step by step.

1. c) Write the quotient polynomial $C(x)$ and clearly state the value of the remainder $R$.

Calculate the quotient and the remainder of the following division using Ruffini's Rule. Pay attention, as the divisor does not have a leading coefficient equal to 1:

Remember that to use Ruffini's rule, the divisor must be of the form $(x - a)$. How could you transform $2x - 1$ to give it that form?

The area of a rectangular padel court is given by the polynomial $A(x) = 2x^2 + 11x + 12$. If the width of the court measures $(x + 4)$ meters, determine the algebraic expression that represents its length.

A logistics company uses cardboard boxes whose volume in cubic centimeters is given by the polynomial $V(x) = 2x^3 + 9x^2 + 7x - 6$. If the base area of the box is $B(x) = 2x^2 + 5x - 3$, calculate the polynomial that represents the height $H(x)$ of the box.

The total profit of a video game store from the sale of a new release is expressed by the polynomial $B(x) = x^3 + 15x^2 + 54x$. If the profit for each video game sold is represented by the binomial $(x + 6)$, perform the division to find the polynomial that represents the total number of video games sold.

Without carrying out long division or using Ruffini's rule, calculate the value of the remainder of the following operation using the Remainder Theorem:

Calculate the value of the letter $k$ so that the division is exact (that is, the remainder is $0$):

Without carrying out the division, use the Remainder Theorem to find out whether the number $x = 2$ is a root of the polynomial $P(x) = 2x^3 - 5x^2 - x + 6$. Justify your answer by indicating whether the division of $P(x)$ by $(x - 2)$ has no remainder (is exact) or not.

Find the value of the parameter $k$ so that the division of the polynomial $P(x) = x^3 - kx^2 + 4x - 8$ by the binomial $(x - 2)$ has a remainder equal to $10$.

Find the values of the parameters $a$ and $b$ so that the polynomial $P(x) = x^4 - ax^3 + bx^2 + x - 6$ is divisible simultaneously by $(x - 1)$ and by $(x + 2)$. Hint: use the Remainder Theorem to set up a system of equations.

Determine the value of $k$ so that the remainder of the division $(2x^4 - 5x^3 + kx^2 - 12) : (x + 2)$ is equal to the remainder of the division $(x^3 + 2x^2 - x + 10) : (x - 3)$.

Check whether the number $a = 3$ is a root of the polynomial $P(x) = x^3 - 3x^2 + x - 3$ by carrying out division using Ruffini's rule and observing the remainder obtained. Justify your answer based on the result of the operation.

Check whether the binomial $(x + 1)$ is a factor of the polynomial $P(x) = x^4 - 1$. To do this, you can use long division, Ruffini's rule, or the Factor Theorem. Explain what it means for a binomial to be a "factor" of another polynomial in terms of the remainder of the division.

Consider the polynomial $P(x) = 3x^3 - kx^2 + 12x - 4$.

1. Calculate the value of $k$ so that $x = 2$ is a root of the polynomial.

2. Once $k$ is found, use Ruffini's Rule to divide $P(x)$ by $(x - 2)$ and check that the remainder is zero.

3. Is the quotient obtained a monic quadratic polynomial? (A polynomial is monic if its leading coefficient is 1).