Exercises


  1. Divide the polynomial \( 2x^3 - 7x^2 - 4x + 36 \) by each of the following divisors. Express the polynomial in terms of the quotient, divisor, and remainder. State any restrictions on the variable.
    1. \( x - 1 \)
    2. \( x + 3 \)
    3. \( 2x - 3 \)
  2. Express each division in terms of the quotient, divisor, and remainder.
    1. \( \dfrac{x^3 + x - 8x^2 + 37}{x + 4}, x \neq -4 \)
    2. \( \dfrac{10x^3 - 9x^2 - 8x + 11}{5x - 2}, x \neq \frac{2}{5} \)
    3. \( \dfrac{3x + 2 - x^3}{x - 2}, x \neq 2 \)
    4. \( \dfrac{x^4 - 2x^2 + 12}{x - 4}, x \neq 4 \)
    5. \( \dfrac{x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18}{x^2 - 2x - 3}, x \neq -1, 3 \)
  3. The polynomial \( 6x^3 - 5x^2 - 49x + 60 \) is divided by \( 2x - 5 \).
    1. Identify the restrictions on \( x \).
    2. Show that the remainder is zero.
    3. Express the polynomial dividend in terms of the divisor, quotient, and remainder.
    4. What conclusion can be drawn when the remainder is zero?
    5. Express the polynomial in fully factored form.
    1. Divide \( f(x) = x^3 + (a + b)x^2 + (ab + c)x + ac \) by \( d(x) = x + a \) and express \( f(x) \) in terms of the divisor, quotient, and remainder.
    2. Using your findings in part a, create a cubic polynomial that has \( x - 2 \) as a factor. Verify your answer by carrying out the division.
    1. When dividing a \( 5^{\text{th}} \) degree polynomial by a \( 2^{\text{nd}} \) degree polynomial, what is the degree of the quotient and the maximum degree of the remainder?
    2. When dividing a \( n^{\text{th}} \) degree polynomial by a divisor of degree \( m \), where \(m\) and \(n\) are positive integers and \(m\leq n\), what is the degree of the quotient and the maximum degree of the remainder?
  6. When a polynomial \( P(x) \) is divided by \( x + 3 \), the quotient is \( 3x^2 - 5x + 4 \) and the remainder is \( -10 \). Find \( P(x) \) in standard form.
  7. Find the divisor given the divident, quotient, and remainder.
    1. The dividend is \( 3x^3 - 5x^2 - 7x - 1 \), the quotient is \( 3x^2 + 4x + 5 \), and the remainder is 14.
    2. The dividend is \( 2x^4 + 11x^3 + 5x^2 - 31x + 7 \), the quotient is \( 2x^2 + 3x - 5 \), and the remainder is \( -8x + 2 \).
  8. The volume of a cylinder is given by \( (\pi x^3 + 4\pi x^2 - 3\pi x - 18 \pi) \text{ cm}^3 \). If the radius of the cylinder is \( (x + 3) \text{ cm} \), determine the height of the cylinder in terms of \( x \).
  9. Given that
    \( \dfrac{x^4 - 3x^3 + px^2 - 11x - 7}{x^2 + 2x + 1} = x^2 + qx - 3 + \dfrac{r}{x^2 + 2x + 1} \) where \( p, q, r \in \mathbb{R} \),
    find \(p, q \), and \( r \).
    1. When a number \( n \) is divided by \( 7 \), the remainder is \( 3 \). What is the remainder when \( 4n \) is divided by \( 7 \)?
    2. When \( P(x) \) is divided by \( (x + 1) \), the remainder is \( 3 \). What is the remainder when \( xP(x) \) is divided by \( (x + 1) \)?
  11. Prove that \( \underbrace{2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^{n - 2} + 2^{n - 1}}_{n \text{ terms}} = 2^n - 1 \), by dividing \( x^n - 1 \) by \( x - 1 \).