Polynomial Division Theorem

Exercise 1 : Polynomial Long Division

Divide using polynomial long division.

  1. $(x^2 + 5x + 6) \div (x + 2)$

  2. $(2x^2 - 7x + 3) \div (x - 3)$

  3. $(x^3 - 8) \div (x - 2)$

  4. $(3x^3 - 2x^2 + 4x - 1) \div (x^2 - x + 1)$

  5. $(x^4 - 16) \div (x^2 + 4)$

  6. $(2x^4 - 3x^3 + x^2 - 5x + 2) \div (x^2 - 2x + 1)$

Exercise 2 : Synthetic Division

Divide using synthetic division.

  1. $(x^3 - 2x^2 - 5x + 6) \div (x - 1)$
  2. $(2x^3 + 3x^2 - 4x + 1) \div (x + 2)$
  3. $(x^4 - 3x^2 + 2x - 5) \div (x - 3)$

Exercise 3 : Remainder Theorem Applications

Use the Remainder Theorem to find the remainder.

  1. $P(x) = 2x^3 - 5x^2 + 3x - 7$ divided by $(x - 2)$
  2. $Q(x) = x^4 - 2x^3 + x - 5$ divided by $(x + 1)$
  3. $R(x) = 3x^5 - 2x^3 + 4x - 1$ divided by $(x - 3)$

Exercise 4 : Factor Theorem Applications

Use the Factor Theorem to determine if the given binomial is a factor.

  1. Is $(x - 1)$ a factor of $x^3 - 3x^2 + 2x - 2$?
  2. Is $(x + 2)$ a factor of $2x^3 + x^2 - 7x + 2$?
  3. Is $(x - 3)$ a factor of $x^4 - 4x^3 + 2x^2 - 5x + 6$?

Exercise 5 : Finding Roots

Find all real roots of the polynomials.

  1. $P(x) = x^3 - 2x^2 - 5x + 6$
  2. $Q(x) = 2x^3 + x^2 - 13x + 6$
  3. $R(x) = x^4 - 5x^2 + 4$

Exercise 6 : Complete Factorization

Completely factor each polynomial.

  1. $x^3 - 8$
  2. $x^4 - 16$
  3. $2x^3 + 3x^2 - 8x + 3$
  4. $x^4 - 3x^3 - 3x^2 + 11x - 6$

Exercise 7 : Rational Root Theorem

Use the Rational Root Theorem to:

  • List all possible rational roots
  • Test each possible root
  • Find all actual roots
  1. $2x^3 - 3x^2 - 11x + 6$
  2. $3x^4 - 4x^3 - 11x^2 + 16x - 4$

Exercise 8 : Polynomial Equations

Solve the polynomial equations.

  1. $x^3 - 3x^2 - 4x + 12 = 0$
  2. $2x^4 - 5x^3 - 3x^2 + 5x - 1 = 0$

Exercise 9 : Word Problems

Solve the following application problems.

  1. The volume of a rectangular box is given by $V(x) = x^3 - 2x^2 - 5x + 6$. If the height is $(x - 1)$, find expressions for the length and width.

  2. When a polynomial $P(x)$ is divided by $(x - 2)$, the remainder is 5. When divided by $(x + 1)$, the remainder is -4. Find the remainder when $P(x)$ is divided by $(x - 2)(x + 1)$.

Exercise 10 : Advanced Problems

Solve these advanced problems.

  1. Find the value of k such that $(x + 2)$ is a factor of $x^3 + kx^2 - 4x - 12$
  2. A polynomial $P(x)$ leaves a remainder of 2 when divided by $(x - 1)$ and a remainder of 1 when divided by $(x - 2)$. What remainder is obtained when $P(x)$ is divided by $(x - 1)(x - 2)$?
  3. Prove that $x^2 + x + 1$ is a factor of $x^{3n+2} + x^{3m+1} + 1$ for all positive integers m and n.

Exercise 11 : Comprehensive Problems

Solve using the most efficient method.

  1. Divide $(3x^4 - 2x^3 + x^2 - 5x + 2) \div (x^2 - x + 1)$
  2. Find all roots of $x^4 - 3x^3 - 6x^2 + 28x - 24 = 0$
  3. Factor completely: $6x^4 - 11x^3 - 35x^2 + 34x + 24$

Exercise 12 : Proof and Justification

Prove or justify the following statements.

  1. Prove the Remainder Theorem.
  2. Explain why synthetic division only works for divisors of the form $(x - c)$.
  3. Show that if a polynomial has real coefficients and a complex root $a + bi$ (where $b \neq 0$), then $a - bi$ must also be a root.