The Remainder Theorem and Synthetic Division

Fundamental Concept of the Remainder Theorem

• The Remainder Theorem provides a specific relationship between a polynomial function and its division by a linear factor.
• If a polynomial function $f(x)$ is divided by a linear factor of the form $(x - c)$, the resulting remainder is equal to the value of the function evaluated at $c$, denoted as $f(c)$.
• This principle holds true whether the division is performed using synthetic division or long division.
• The theorem is practically useful because evaluating a function through synthetic division is often significantly faster and less prone to calculation errors than direct substitution, especially for higher-degree polynomials.

Example 1: Evaluating $f(x) = 2x^3 - 5x^2 + 6x - 12$ at $x = 4$

Method 1: Direct Substitution (The "Old-Fashioned Way")

• The goal is to evaluate $f(4)$ by replacing every instance of $x$ with the number $4$
  • Equation: $f(4) = 2(4)^3 - 5(4)^2 + 6(4) - 12$
• Step-by-step calculation:
  • Evaluate $4^3$: $4 \times 4 = 16$ and $16 \times 4 = 64$
  • Evaluate $4^2$: $4 \times 4 = 16$
  • Multiply terms:
    • $2 \times 64 = 128$
    • $5 \times 16$: Calculated as $(5 \times 10) + (5 \times 6) = 50 + 30 = 80$
    • $6 \times 4 = 24$
  • Combine results:
    • $f(4) = 128 - 80 + 24 - 12$
    • The speaker simplifies $24 - 12 = 12$
    • $128 - 80 = 48$ (derived from $12 - 8 = 4$ and $8 - 0 = 8$)
    • $48 + 12 = 60$
  • Final result: $f(4) = 60$

Method 2: Synthetic Division Confirmation

• Coefficients are extracted from the polynomial: $2, -5, 6, -12$.
• The divisor/evaluative point is $4$.
• Procedure (Multiply, Add, Repeat):
  1. Bring down the first coefficient: $2$.
  2. Multiply $4 \times 2 = 8$.
  3. Add $-5 + 8 = 3$.
  4. Multiply $4 \times 3 = 12$.
  5. Add $6 + 12 = 18$.
  6. Multiply $4 \times 18$: Calculated as $(4 \times 10) + (4 \times 8) = 40 + 32 = 72$
  7. Add the final constant term: $-12 + 72 = 60$.
• Conclusion: The remainder is $60$, which matches $f(4)$, confirming the theorem.

Example 2: Polynomial with Missing Terms and Placeholders

• Given function: $f(x) = 3x^4 - 7x^3 - 9x + 12$.
• Goal: Evaluate $f(5)$.
• Crucial Step: Notice there is no $x^2$ term in the expression. To perform synthetic division, a placeholder of $0x^2$ must be inserted.

Synthetic Division Process

• Coefficients used: $3, -7, 0, -9, 12$.
• Divisor: $5$.
• Steps:
  1. Bring down $3$.
  2. $5 \times 3 = 15$.
  3. $-7 + 15 = 8$.
  4. $5 \times 8 = 40$.
  5. $0 + 40 = 40$.
  6. $5 \times 40 = 200$.
  7. $-9 + 200 = 191$.
  8. $5 \times 191$: Calculated as $5 \times (200 - 9) = 1000 - 45 = 955$.
  9. $12 + 955 = 967$.
• Remainder: $967$. Therefore, $f(5) = 967$.

Confirmation via Substitution

• Expression: $f(5) = 3(5)^4 - 7(5)^3 - 9(5) + 12$
• Powers of 5:
  • $5^1 = 5$
  • $5^2 = 25$
  • $5^3 = 125$
  • $5^4 = 625$
• Calculations:
  • $3 \times 625$: Calculated as $(3 \times 600) + (3 \times 25) = 1800 + 75 = 1875$
  • $7 \times 125$: Calculated via the "seven quarters" metaphor as $700 + 175 = 875$
  • Subtraction: $1875 - 875 = 1000$
  • Final steps: $1000 - 45 + 12 = 955 + 12 = 967$.

Example 3: evaluating $2x^4 - x^2 + 30$ at $x = 3$

• The speaker evaluates the expression at $x = 3$ using synthetic division first.

Synthetic Division Execution

• Coefficients identified (including placeholders for $x^3$ and $x^1$): $2, 0, -3, 0, 30$.
• Note: While the spoken equation is $2x^4 - x^2 + 30$, the speaker uses $-3$ as the coefficient for $x^2$ during the demonstration.
• Steps:
  1. Bring down $2$: Result is $2$.
  2. $3 \times 2 = 6$; $0 + 6 = 6$.
  3. $3 \times 6 = 18$; $-3 + 18 = 15$.
  4. $3 \times 15 = 45$; $0 + 45 = 45$.
  5. $3 \times 45$: Calculated as $(3 \times 40) + (3 \times 5) = 120 + 15 = 135$.
  6. $135 + 30 = 165$.
• Remainder: $165$.

Checking the Answer

• Substitution expression used: $f(3) = 2(3)^4 - 3(3)^2 + 13$
  • Note: There is a discrepancy between the stated equation and the check, but calculations proceed based on the synth division result.
• Evaluate powers: $3^4 = 81$ ($3 \times 3 \times 3 \times 3$ is $9 \times 9$) and $3^2 = 9$.
• Calculations:
  • $2 \times 81 = 162$
  • $3 \times 9 = 27$
  • The speaker combines $-27 + 30 = 3$ (correcting the constant term to $30$ during analysis).
  • $162 + 3 = 165$.

Instructional Interaction and Encouragement

• Throughout the tutorial, the speaker encourages active participation, suggesting viewers "feel free to pause the video" to attempt the synthetic division independently before viewing the solution.
• The speaker emphasizes that synthetic division is a reliable shortcut for finding values of complex functions at specific points without lengthy algebraic expansions.