Algebra 101 - Unit 5

Lesson 28: Dividing Polynomials (Long Division, Synthetic Division)

Dividing polynomials can be done using long division or synthetic division (for divisors of the form x − c). This process is similar to numerical division but applied to algebraic expressions.

Long Division Method

Steps:

1. Arrange both dividend and divisor in descending powers of x.
2. Divide the first term of the dividend by the first term of the divisor; this is the first term of the quotient.
3. Multiply the entire divisor by this term and subtract the result from the dividend.
4. Bring down the next term and repeat until all terms are used.

Example:

Divide (2x³ + 3x² − x + 5) ÷ (x + 2)

1. 2x³ ÷ x = 2x² → first term of quotient
2. Multiply: 2x²*(x + 2) = 2x³ + 4x²
3. Subtract: (2x³ + 3x² − x + 5) − (2x³ + 4x²) = −x² − x + 5
4. −x² ÷ x = −x → next term
5. Multiply: −x*(x + 2) = −x² − 2x
6. Subtract: (−x² − x + 5) − (−x² − 2x) = x + 5
7. x ÷ x = 1 → next term
8. Multiply: 1*(x + 2) = x + 2
9. Subtract: (x + 5) − (x + 2) = 3 → remainder

Answer: 2x² − x + 1 + 3/(x + 2)

Synthetic Division Method

Use synthetic division when dividing by x − c.

Example: Divide 2x³ + 3x² − x + 5 by x + 2 (note: x + 2 = x − (−2))

1. Write coefficients: 2, 3, −1, 5
2. c = −2
3. Bring down the first coefficient: 2
4. Multiply 2*−2 = −4, add to next coefficient: 3 + (−4) = −1
5. Multiply −1*−2 = 2, add: −1 + 2 = 1
6. Multiply 1*−2 = −2, add: 5 + (−2) = 3 → remainder

Answer: 2x² − x + 1 + 3/(x + 2)

Practice Problems

1. Divide x³ − 6x² + 11x − 6 by x − 1 (use synthetic division)
2. Divide 3x³ + 5x² − x + 2 by x + 1 (use long division)
3. Divide 2x⁴ − 3x³ + x² − 5 by x − 2 (synthetic)
4. Divide x³ + 4x² − x − 4 by x + 2 (long division)