Chapter 09 | Division of Polynomials | First Year Math

 

Chapter 09 | Division of Polynomials | First Year Math

The Division of Polynomials is an important topic in algebra. It helps to simplify expressions and solve equations easily.

What is a Polynomial?

A polynomial is an expression with numbers and powers of x added, subtracted, or multiplied.

Example:
2x^3 + 3x^2 - x + 5

• 2, 3, -1, 5 are called coefficients
• x is the variable
• The highest power of x (here 3) is called the degree

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What is Division of Polynomials?

Division of polynomials means splitting one polynomial (dividend) by another (divisor) to get:

Dividend = (Divisor × Quotient) + Remainder

• Quotient = the result of division
• Remainder = the leftover (degree smaller than divisor)

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Methods of Division

1. Long Division

Steps:

1. Arrange dividend and divisor in descending powers of x.
2. Divide the first term of dividend by the first term of divisor → first term of quotient
3. Multiply the divisor by this term and subtract from dividend
4. Bring down next term and repeat until remainder is smaller than divisor

Example: Divide 2x^3 + 3x^2 - x + 5 by x + 2

1. 2x^3 ÷ x = 2x^2 → first term of quotient
2. Multiply: (x + 2) × 2x^2 = 2x^3 + 4x^2
3. Subtract: (2x^3 + 3x^2) - (2x^3 + 4x^2) = -x^2
4. Bring down -x → -x^2 - x
5. Divide: -x^2 ÷ x = -x
6. Multiply: (x + 2) × -x = -x^2 - 2x
7. Subtract: (-x^2 - x) - (-x^2 - 2x) = x
8. Bring down +5 → x + 5
9. Divide: x ÷ x = 1
10. Multiply: (x + 2) × 1 = x + 2
11. Subtract: (x + 5) - (x + 2) = 3

Quotient = 2x^2 - x + 1
Remainder = 3

Answer: 2x^3 + 3x^2 - x + 5 = (x + 2)(2x^2 - x + 1) + 3

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2. Synthetic Division

Use this only if divisor is x - c.

Steps:

1. Write coefficients of dividend
2. Bring down first coefficient
3. Multiply by c (from x - c) and write under next coefficient
4. Add column and repeat
5. Last number = remainder, others = coefficients of quotient

Example: Divide 2x^3 + 3x^2 - x + 5 by x - 1

Coefficients: 2, 3, -1, 5
c = 1

Step by step:

• Bring down 2 → 2
• Multiply 2 × 1 = 2 → Add to next coefficient: 3 + 2 = 5
• Multiply 5 × 1 = 5 → Add to next coefficient: -1 + 5 = 4
• Multiply 4 × 1 = 4 → Add to next coefficient: 5 + 4 = 9

Quotient = 2x^2 + 5x + 4
Remainder = 9

Answer: 2x^3 + 3x^2 - x + 5 = (x - 1)(2x^2 + 5x + 4) + 9

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PDF SOLUTION:

Rules and Shortcuts

1. Division Algorithm: Dividend = Divisor × Quotient + Remainder
2. Remainder Theorem: Remainder of P(x) ÷ (x - a) = P(a)
3. Factor Theorem: If P(a) = 0, then x - a is a factor

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Tips

• Always write terms in descending powers of x
• If a term is missing, write 0 as its coefficient
• Use synthetic division for x - c (faster)
• Use long division for any divisor
• Remainder degree < divisor degree

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Practice Examples

1. Divide x^4 - 2x^3 + 3x - 5 by x - 1
2. Divide 3x^3 + 4x^2 - x + 6 by x + 2
3. Use synthetic division to divide 2x^3 + 5x^2 - x + 1 by x - 1

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Uses of Polynomial Division

• Solve polynomial equations
• Simplify algebra expressions
• Prepare for calculus
• Find graph asymptotes in rational functions

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Summary

• Polynomials are expressions with powers of x
• Division gives quotient and remainder
• Long division works for all divisors, synthetic division is faster for x - c
• Remainder and factor theorems help check answers
• Practice makes division easy and fast