Division of polynomials

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 $3x$ metres, what is the algebraic expression that represents the length of the studio?

Given the polynomial division $(6x^4 - x^3 - 2x^2 + 8x - 1) : (2x^2 - x + 1)$:

1. a) Before carrying out the algorithm, estimate and reason:

1. What exact degree will the quotient polynomial have?

2. What is the maximum degree the remainder can have? Could the remainder be a second-degree polynomial like $x^2 - 3$? Justify it according to the definition of polynomial division.

3. What will be the leading monomial (the term with the highest degree) of the quotient?

1. b) Perform the traditional long division using the box method algorithm and check whether your estimates about the degrees and the leading monomial match the obtained result.

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:

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

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 $(x + 4)$ metres, determine the algebraic expression that represents its length.

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: $D(x) = d(x) \times C(x) + R(x)$.

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 logistics company uses cardboard boxes whose volume in cubic centimetres 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.

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 $R$.

Aoife wants to calculate the quotient and remainder of the division $(2x^3 - 7x + 5) : (x - 3)$ using Ruffini's rule. In her notebook, she has set up the table by writing the numbers $2$, $-7$, $5$ in the top row and, on the left, as the value associated with the divisor, the number $-3$.

1. a) Identify and explain the two conceptual mistakes Aoife made before starting to operate: one concerning the sign of the divisor and the other concerning the coefficients of the dividend.
2. b) Correct both mistakes by properly redoing Ruffini's table, and explicitly write the obtained quotient polynomial $C(x)$ and 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?

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 $10$.

Consider the polynomial $P(x) = x^4 - 3x^3 + 2x - 15$, and suppose you want to study its division by $(x - 3)$.

1. a) If the sole purpose of an exercise were to find out whether the division is exact (that is, if the remainder is zero), compare two possible approaches: Ruffini's rule and the Remainder Theorem. Justify which of the two procedures is faster and more efficient to answer that question.
2. b) Apply the method you selected as the most efficient in the previous section, calculate the remainder, and deduce with justification whether $(x - 3)$ is a factor of $P(x)$ or not.
3. c) If you were now required to find the quotient polynomial $C(x)$, reason whether the Remainder Theorem method would still be useful to you and obtain that quotient using the appropriate method.

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.

Consider a polynomial $P(x)$ whose full expression is unknown, but we know two results of its divisions: when dividing $P(x)$ by $(x - 2)$, the remainder obtained is $7$, and when dividing $P(x)$ by $(x + 1)$, the remainder obtained is $-2$.

1. a) If we divide $P(x)$ by the second-degree polynomial $d(x) = (x - 2) \times (x + 1) = x^2 - x - 2$, reason in detail from the relationship between degrees why the remainder $R(x)$ must have at most degree $1$ and can always be written as $R(x) = mx + n$.
2. b) State the fundamental division identity: $P(x) = (x^2 - x - 2) \times C(x) + (mx + n)$. Without knowing the quotient $C(x)$, substitute the appropriate values of $x$ into this identity to formulate a system of two linear equations in the unknowns $m$ and $n$.
3. c) Solve the system and write the final algebraic expression of the remainder $R(x)$.

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.

We want to determine a third-degree polynomial $P(x)$ whose leading coefficient is $1$. The following information about its divisions is known:

1. When dividing it by $(x - 2)$, the division is exact.

2. When dividing it by $(x + 1)$, the remainder is $0$.

3. Its constant term is $-8$.

1. a) Using the first two conditions, determine two roots of $P(x)$ and write its expression in factored form: $P(x) = (x - 2) \times (x + 1) \times (x - a)$, where $a$ is an unknown root.
2. b) From the constant term, set up a simple equation to calculate the exact value of the root $a$ and expand the polynomial to express it in the standard form $P(x) = x^3 + Bx^2 + Cx + D$.
3. c) Use the Remainder Theorem (without doing long division or Ruffini's rule) to check whether the binomial $(x + 4)$ is also a factor of $P(x)$.

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.