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How to Expand Polynomials Step by Step?

Published in Polynomial Expansion 2 mins read

Expanding polynomials involves removing parentheses and combining like terms to simplify the expression. Here's a step-by-step guide:

Understanding Polynomial Expansion

A polynomial is considered expanded when:

  • No variable appears within parentheses.
  • All like terms have been combined.

According to the provided reference, to expand a polynomial, you must multiply its factors (often by using the distributive property) or perform the indicated operations, then combine all like terms.

Steps to Expand Polynomials

Here’s a detailed breakdown:

  1. Identify the Operations: Determine the operations that need to be performed (multiplication, addition, subtraction).

  2. Apply the Distributive Property (if needed): If the polynomial involves multiplication of a term with a group inside parentheses, use the distributive property. This involves multiplying the term outside the parentheses by each term inside.

    • Example: a(b + c) = ab + ac

  3. Multiply Polynomials (if needed): If you're multiplying two polynomials (e.g., (a + b)(c + d)), multiply each term in the first polynomial by each term in the second polynomial.

    • Example: (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd

  4. Simplify: Remove all parentheses by performing the multiplications.

  5. Combine Like Terms: Identify terms with the same variable and exponent, then combine their coefficients.

    • Example: 3x² + 2x² = 5x²

Examples

Example 1: Simple Distribution

Expand 2x(x + 3)

  1. Distribute: 2x * x + 2x * 3
  2. Multiply: 2x² + 6x
  3. Combine Like Terms: There are no like terms to combine.
  4. Result: 2x² + 6x

Example 2: Multiplying Binomials

Expand (x + 2)(x - 3)

  1. Multiply: x(x - 3) + 2(x - 3)
  2. Distribute: x² - 3x + 2x - 6
  3. Combine Like Terms: -3x + 2x = -x
  4. Result: x² - x - 6

Example 3: Combining Distribution and Like Terms

Expand 3(y - 1) + 2(y + 4)

  1. Distribute: 3y - 3 + 2y + 8
  2. Combine Like Terms: 3y + 2y = 5y and -3 + 8 = 5
  3. Result: 5y + 5

Tips for Success

  • Be careful with signs: Pay close attention to negative signs when distributing.
  • Double-check: Review your work to ensure you've multiplied and combined terms correctly.
  • Practice: The more you practice, the easier it will become.

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