The Remainder Theorem - Algebra-2

1. Fundamental Concepts

Definition: The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - c$, the remainder is $f(c)$.
Synthetic Division: A simplified method for dividing polynomials by binomials of the form $x - c$.
Application: Used to evaluate polynomials and find roots.

2. Key Concepts

Theorem Statement: $$\text{If } f(x) \text{ is divided by } x - c, \text{ then the remainder is } f(c).$$

Synthetic Division Process: List coefficients of $f(x)$, bring down the leading coefficient, multiply by $c$ and add to the next coefficient, repeat until all terms are processed.

Root Finding: Use synthetic division to test potential roots and confirm with the Remainder Theorem.

3. Examples

Example 1 (Basic)

Problem: Use synthetic division to divide $f(x) = 2x^3 + 3x^2 - 11x + 6$ by $x - 2$.

Step-by-Step Solution:

Write down the coefficients: $2, 3, -11, 6$.
Set up synthetic division with $c = 2$.

              2 | 2  3  -11  6                 |   4  14  6              -----------------                  2  7   3  12

The quotient is $2x^2 + 7x + 3$ and the remainder is $12$.

Validation: Substitute $x = 2$ into $f(x)$: $2(2)^3 + 3(2)^2 - 11(2) + 6 = 16 + 12 - 22 + 6 = 12$. The remainder matches.

Example 2 (Intermediate)

Problem: Determine if $x = -3$ is a root of $g(x) = x^4 + 2x^3 - 7x^2 - 18x + 27$.

Step-by-Step Solution:

Write down the coefficients: $1, 2, -7, -18, 27$.
Set up synthetic division with $c = -3$.

              -3 | 1  2  -7  -18  27                   |   -3  3  12  18              ----------------------                  1 -1  -4  -6  0

The remainder is $0$, so $x = -3$ is a root.

Validation: Substitute $x = -3$ into $g(x)$: $(-3)^4 + 2(-3)^3 - 7(-3)^2 - 18(-3) + 27 = 81 - 54 - 63 + 54 + 27 = 0$. The remainder matches.

4. Problem-Solving Techniques

Check Coefficients: Ensure all coefficients are correctly listed before starting synthetic division.
Vertical Alignment: Align coefficients vertically to avoid errors in addition and multiplication.
Remainder Verification: Always verify the remainder by substituting $c$ back into the original polynomial.