Polynomial Operations- Following the Correct Order

What Are Polynomials Anyway?

A polynomial is an expression with multiple terms made up of constants, variables, and exponents. The word "poly" means many, and "nomial" means terms. So you're dealing with expressions that have multiple terms combined through addition, subtraction, or multiplication.

Examples include:

The degree of a polynomial is the highest exponent. In 3x² + 2x - 5, the degree is 2 because x² is the highest power.

The Order of Operations: This Is Non-Negotiable

Here's the thing about math — the order you do operations matters. Get it wrong and you'll end up with the wrong answer every single time. There's no "close enough" in math.

PEMDAS/BODMAS Explained

Use this sequence every time, no exceptions:

  1. Parentheses or Brackets — solve anything inside first
  2. Exponents or Orders — powers and roots
  3. Multiplication and Division — left to right
  4. Addition and Subtraction — left to right

Think of it as a chain of command. Each level only acts after the level above it is completely finished.

Adding and Subtracting Polynomials

This one's straightforward — combine like terms only. Like terms have the same variable raised to the same power.

How To Add Polynomials

Example: (3x² + 2x - 5) + (x² - 4x + 3)

Step 1: Remove parentheses

3x² + 2x - 5 + x² - 4x + 3

Step 2: Group like terms

3x² + x² + 2x - 4x - 5 + 3

Step 3: Combine each group

4x² - 2x - 2

That's it. Just make sure you're adding the coefficients correctly and keeping the exponents exactly the same.

How To Subtract Polynomials

Subtraction requires extra care. Distribute the negative sign to every term in the second polynomial before combining.

Example: (5x³ + 3x² - 2x) - (2x³ - 4x² + 6x)

Step 1: Distribute the negative

5x³ + 3x² - 2x - 2x³ + 4x² - 6x

Step 2: Combine like terms

3x³ + 7x² - 8x

The most common mistake here is forgetting to change the signs. If you skip distributing the negative, you'll get it wrong every time.

Multiplying Polynomials

This is where things get more complex. You need to multiply every term in the first polynomial by every term in the second polynomial.

Multiplying a Monomial by a Polynomial

Example: 3x²(2x³ + 5x - 4)

Multiply 3x² by each term:

3x² × 2x³ = 6x⁵

3x² × 5x = 15x³

3x² × (-4) = -12x²

Answer: 6x⁵ + 15x³ - 12x²

When multiplying powers with the same base, add the exponents. x² × x³ = x⁵.

Multiplying Two Binomials (FOIL Method)

For (a + b)(c + d), use FOIL:

Example: (x + 3)(x + 5)

Combine: x² + 5x + 3x + 15 = x² + 8x + 15

Then combine the like terms (5x + 3x = 8x).

Multiplying Larger Polynomials

For polynomials with more than two terms, use the distributive property systematically. Multiply each term of the first polynomial by each term of the second.

Example: (x + 2)(x² + 3x - 4)

Combine: x³ + 3x² + 2x² - 4x + 6x - 8 = x³ + 5x² + 2x - 8

Dividing Polynomials

Polynomial division comes in two flavors: long division and synthetic division. Synthetic is faster but only works with specific conditions.

Polynomial Long Division

Works like regular long division but with variables.

Example: Divide x² + 5x + 6 by x + 2

  1. Divide the first term: x² ÷ x = x
  2. Multiply: x(x + 2) = x² + 2x
  3. Subtract: (x² + 5x) - (x² + 2x) = 3x
  4. Bring down: 3x + 6
  5. Divide: 3x ÷ x = 3
  6. Multiply: 3(x + 2) = 3x + 6
  7. Subtract: (3x + 6) - (3x + 6) = 0

Answer: x + 3

Synthetic Division

Use this when dividing by a linear binomial in the form (x - c). It's faster and requires less writing.

Example: Divide x² + 5x + 6 by (x + 2), which is (x - (-2))

  1. Write the coefficients: 1, 5, 6
  2. Use -2 (the value that makes the divisor zero)
  3. Bring down 1
  4. Multiply -2 × 1 = -2, write under 5
  5. Add: 5 + (-2) = 3
  6. Multiply -2 × 3 = -6, write under 6
  7. Add: 6 + (-6) = 0

Result: x + 3

Synthetic division only works with linear divisors where the coefficient of x is 1. For anything else, use long division.

Operations Comparison Table

Operation Key Rule Common Mistake
Addition Combine like terms only Adding unlike terms together
Subtraction Distribute negative sign first Forgetting to change signs
Multiplication Multiply every term by every term Missing terms during distribution
Division Use long or synthetic division Using synthetic when conditions don't allow

Practical Getting Started Checklist

Before you start any polynomial problem:

The Bottom Line

Polynomial operations aren't complicated — they're just step-by-step procedures. The mistakes people make usually come from skipping steps or not following the order of operations correctly.

For addition and subtraction: combine like terms only. For multiplication: distribute everything. For division: choose the right method based on your divisor. Get these fundamentals right and polynomial operations become routine.