Dividing Polynomials Using Long Division and Synthetic Division

Fundamentals of Polynomial Division

• Polynomial division is an algebraic technique used to divide a polynomial dividend $P(d)$ by a polynomial divisor $D(d)$, yielding a quotient polynomial $Q(d)$ and a remainder polynomial $R(d)$.

• The division algorithm for polynomials states that for any polynomial dividend $P(d)$ and non-zero polynomial divisor $D(d)$, there exist unique polynomials $Q(d)$ and $R(d)$ such that:   $P(d) = D(d) \times Q(d) + R(d)$

• Expressed as a rational fraction:   $\frac{P(d)}{D(d)} = Q(d) + \frac{R(d)}{D(d)}$

• The degree of the remainder polynomial $R(d)$ must strictly be less than the degree of the divisor polynomial $D(d)$.

• If $R(d) = 0$, then $D(d)$ divides $P(d)$ evenly without a remainder, meaning $D(d)$ and $Q(d)$ are exact factors of $P(d)$.

• If $R(d) \neq 0$, the remainder is expressed as a proper fraction with the divisor as the denominator: $\frac{R(d)}{D(d)}$.

Step-by-Step Procedure for Polynomial Long Division

• Step 1: Standard Form and Missing Terms Setup

  • Write both the dividend $P(d)$ and divisor $D(d)$ in descending order of exponents (standard form).
  • Insert placeholder terms with a coefficient of $0$ for any missing power of the variable (e.g., $0d^0$ if a constant term is absent).

• Step 2: Leading Term Division

  • Divide the leading term of the current dividend by the leading term of the divisor to determine the first/next term of the quotient $Q(d)$.

• Step 3: Multiplication

  • Multiply the newly obtained term of the quotient by the entire divisor polynomial.

• Step 4: Subtraction

  • Subtract the result from the current dividend terms. Ensure signs of all terms being subtracted are correctly distributed and inverted.

• Step 5: Bring Down Next Term

  • Bring down the next highest power term from the original dividend.

• Step 6: Iteration and Completion

  • Repeat steps 2 through 5 until the degree of the remaining expression is strictly less than the degree of the divisor polynomial.

Worked Example: Polynomial Long Division

• Problem Statement:$\text{Divide: } (2d^4 + 2d^3 - 18d^2 - 18d) \div (d + 3)$   If the polynomial does not divide evenly, include the remainder as a fraction.

• Dividend: $2d^4 + 2d^3 - 18d^2 - 18d + 0$

• Divisor: $d + 3$

• Detailed Step-by-Step Solution:

  • First Term Computation:

  • Divide the leading term $2d^4$ by $d$:       $\frac{2d^4}{d} = 2d^3$

  • Multiply $2d^3$ by $(d + 3)$:       $2d^3 \times (d + 3) = 2d^4 + 6d^3$

  • Subtract $(2d^4 + 6d^3)$ from $(2d^4 + 2d^3)$:       $(2d^4 + 2d^3) - (2d^4 + 6d^3) = -4d^3$

  • Bring down $-18d^2$ to form the new expression:       $-4d^3 - 18d^2$

  • Second Term Computation:

  • Divide the leading term $-4d^3$ by $d$:       $\frac{-4d^3}{d} = -4d^2$

  • Multiply $-4d^2$ by $(d + 3)$:       $-4d^2 \times (d + 3) = -4d^3 - 12d^2$

  • Subtract $(-4d^3 - 12d^2)$ from $(-4d^3 - 18d^2)$:       $(-4d^3 - 18d^2) - (-4d^3 - 12d^2) = -6d^2$

  • Bring down $-18d$ to form the new expression:       $-6d^2 - 18d$

  • Third Term Computation:

  • Divide the leading term $-6d^2$ by $d$:       $\frac{-6d^2}{d} = -6d$

  • Multiply $-6d$ by $(d + 3)$:       $-6d \times (d + 3) = -6d^2 - 18d$

  • Subtract $(-6d^2 - 18d)$ from $(-6d^2 - 18d)$:       $(-6d^2 - 18d) - (-6d^2 - 18d) = 0$

  • Bring down the constant term $0$:       $\text{Remainder } R = 0$

• Final Answer:$2d^3 - 4d^2 - 6d$

Synthetic Division Alternative Method

• Synthetic division is a simplified shortcut for dividing polynomials when the divisor is a linear factor of the form $(d - c)$.

• For the linear divisor $d + 3$, set $d + 3 = 0 \rightarrow c = -3$

• Set up the coefficients of the dividend $2d^4 + 2d^3 - 18d^2 - 18d + 0$ in sequence:   $\text{Coefficients: } 2, \, 2, \, -18, \, -18, \, 0$

• Execution Steps:

  • Drop the leading coefficient $2$ to the bottom row.
  • Multiply $2$ by $-3$ to get $-6$, and write it under the second coefficient $2$.
  • Add $2 + (-6) = -4$
  • Multiply $-4$ by $-3$ to get $12$, and write it under $-18$.
  • Add $-18 + 12 = -6$
  • Multiply $-6$ by $-3$ to get $18$, and write it under $-18$.
  • Add $-18 + 18 = 0$
  • Multiply $0$ by $-3$ to get $0$, and write it under the constant $0$
  • Add $0 + 0 = 0$

• Resulting Synthetic Division Table Row:$\text{Quotient Coefficients: } 2, \, -4, \, -6, \, 0 \quad \text{with Remainder } 0$

• Translating back to polynomial form with degree reduced by 1 ($4 - 1 = 3$):   $Q(d) = 2d^3 - 4d^2 - 6d$

Verification of Polynomial Division

• To verify that a quotient $Q(d)$ is correct, multiply the quotient by the divisor $D(d)$ and add the remainder $R(d)$:   $D(d) \times Q(d) + R(d) = (d + 3)(2d^3 - 4d^2 - 6d) + 0$

• Expand the product using the distributive property:   $(d + 3)(2d^3 - 4d^2 - 6d) = d(2d^3 - 4d^2 - 6d) + 3(2d^3 - 4d^2 - 6d)$$= (2d^4 - 4d^3 - 6d^2) + (6d^3 - 12d^2 - 18d)$

• Combine like terms:   $= 2d^4 + (-4d^3 + 6d^3) + (-6d^2 - 12d^2) - 18d$$= 2d^4 + 2d^3 - 18d^2 - 18d$

• The expanded polynomial matches the original dividend exactly, confirming the solution is accurate.