Polynomial Division: Synthetic Division in College Algebra

Polynomial and Rational Functions

Synthetic Division of Polynomials

Synthetic division is a simplified method for dividing a polynomial by a linear divisor of the form x - c. It is especially useful for quickly finding the quotient and remainder when dividing by such divisors.

• Key Concept: Synthetic division can only be used when the divisor is linear (degree 1) and has a leading coefficient of 1.

• Purpose: To divide a polynomial by a binomial of the form x - c efficiently.

Steps for Synthetic Division

1. Write the coefficients of the dividend polynomial in descending order of degree. If any degree is missing, use 0 as its coefficient.

2. Write the zero of the divisor (c from x - c) to the left.

3. Bring down the leading coefficient.

4. Multiply the zero by the number just written below the line, and write the result in the next column.

5. Add the numbers in the column and write the sum below the line.

6. Repeat the multiply and add steps for all columns.

7. The final row gives the coefficients of the quotient polynomial, with the last number as the remainder.

Example

Divide $-7x^3 + x^2 - 5x^2 + 9$ by $x + 1$ using synthetic division.

• Step 1: Arrange the polynomial in standard form and combine like terms: $-7x^3 - 4x^2 + 9$

• Step 2: The divisor is $x + 1$, so $c = -1$.

• Step 3: Write the coefficients: $-7$, $-4$, $0$, $9$ (note: $0$for the missing$x$ term).

• Step 4: Set up the synthetic division:

-1
	-7	-4	0	9
	-7	3	-3	12

• Step 5: The bottom row (except the last value) gives the coefficients of the quotient. The last value is the remainder.

• Quotient: $-7x^2 + 3x - 3$

• Remainder: $12$

Final Answer: $\displaystyle \frac{-7x^3 - 4x^2 + 9}{x + 1} = -7x^2 + 3x - 3 + \frac{12}{x + 1}$

Additional info: The original question is a typical College Algebra exam question on synthetic division, which is a key skill in polynomial and rational function analysis.