Synthetic Division - Algebra-2

1. Fundamental Concepts

Definition: Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form $$x - c$$.
Process: It involves using only the coefficients of the polynomial and the constant $$c$$ from the divisor $$x - c$$.
Advantages: Faster and less error-prone than long division for certain types of polynomials.

2. Key Concepts

Basic Rule: $$\text{If } f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \text{ then synthetic division can be used to divide } f(x) \text{ by } x - c.$$

Degree Preservation: The degree of the quotient is one less than the degree of the dividend.

Application: Used to find roots of polynomials and simplify expressions in algebra.

3. Examples

Example 1 (Basic)

Problem: Divide $$f(x) = 2x^3 - 3x^2 + 4x - 5$$ by $$x - 2$$.

Step-by-Step Solution:

Write down the coefficients of $$f(x)$$: $$2, -3, 4, -5$$.
Use $$2$$ as the divisor since we are dividing by $$x - 2$$.
Bring down the first coefficient: $$2$$.
Multiply $$2$$ by $$2$$ and add to the next coefficient: $$-3 + 4 = 1$$.
Multiply $$2$$ by $$1$$ and add to the next coefficient: $$4 + 2 = 6$$.
Multiply $$2$$ by $$6$$ and add to the last coefficient: $$-5 + 12 = 7$$.

            2 | 2 -3 4 -5              |   4 2 12            -------------              2 1 6 7

The quotient is $$2x^2 + x + 6$$ and the remainder is $$7$$.

Validation: Substitute $$x = 2$$ into the original polynomial: $$2(2)^3 - 3(2)^2 + 4(2) - 5 = 16 - 12 + 8 - 5 = 7$$. This matches the remainder, confirming the solution.

Example 2 (Intermediate)

Problem: Divide $$g(x) = 4x^4 - 5x^3 + 2x^2 - 3x + 1$$ by $$x + 1$$.

Step-by-Step Solution:

Write down the coefficients of $$g(x)$$: $$4, -5, 2, -3, 1$$.
Use $$-1$$ as the divisor since we are dividing by $$x + 1$$.
Bring down the first coefficient: $$4$$.
Multiply $$4$$ by $$-1$$ and add to the next coefficient: $$-5 - 4 = -9$$.
Multiply $$-1$$ by $$-9$$ and add to the next coefficient: $$2 + 9 = 11$$.
Multiply $$-1$$ by $$11$$ and add to the next coefficient: $$-3 - 11 = -14$$.
Multiply $$-1$$ by $$-14$$ and add to the last coefficient: $$1 + 14 = 15$$.

            -1 | 4 -5 2 -3 1               |   -4 9 -11 14            ---------------              4 -9 11 -14 15

The quotient is $$4x^3 - 9x^2 + 11x - 14$$ and the remainder is $$15$$.

Validation: Substitute $$x = -1$$ into the original polynomial: $$4(-1)^4 - 5(-1)^3 + 2(-1)^2 - 3(-1) + 1 = 4 + 5 + 2 + 3 + 1 = 15$$. This matches the remainder, confirming the solution.

4. Problem-Solving Techniques

Visual Strategy: Use a table to organize the coefficients and the process of synthetic division.
Error-Proofing: Double-check each step by substituting the divisor back into the polynomial to verify the remainder.
Concept Reinforcement: Practice with various degrees of polynomials and divisors to reinforce understanding.