Polynomial Operations

Polynomial Operations

Addition and Subtraction

• To add/subtract polynomials, combine like terms:
  • Add/subtract coefficients of terms with the same variables and exponents
  • Keep the variables and exponents the same
  • Simplify the resulting polynomial

Multiplication

• To multiply polynomials, use the distributive property:
  • Multiply each term in one polynomial by each term in the other polynomial
  • Combine like terms
  • Simplify the resulting polynomial

Polynomial Division

Long Division

• Divide a polynomial by another polynomial using the following steps:
  1. Write the dividend (dividend) and divisor (divisor) in descending order of degree
  2. Divide the leading term of the dividend by the leading term of the divisor
  3. Multiply the divisor by the result and subtract from the dividend
  4. Repeat steps 2-3 until the remainder is 0 or has a lower degree than the divisor
  5. The quotient is the result of the division, and the remainder is the left-over term

Synthetic Division

• A shortcut for dividing a polynomial by a linear divisor (x - c):
  1. Write the coefficients of the dividend in a table, with the leading coefficient on the left
  2. Bring down the leading coefficient
  3. Multiply the divisor's coefficient (c) by the brought-down coefficient, and add to the next column
  4. Repeat step 3 until the last column is reached
  5. The last column represents the quotient and remainder

Factoring

• Factoring a polynomial involves expressing it as a product of simpler polynomials
• Common techniques include:
  • Factoring out greatest common factors (GCFs)
  • Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b))
  • Using the sum and difference formulas (a^2 + 2ab + b^2 = (a + b)^2, a^2 - 2ab + b^2 = (a - b)^2)
  • Factoring by grouping

Polynomial Operations

Addition and Subtraction

• Combine like terms by adding/subtracting coefficients of terms with the same variables and exponents
• Keep variables and exponents the same after combining like terms
• Simplify the resulting polynomial

Multiplication

• Multiply polynomials using the distributive property
• Multiply each term in one polynomial by each term in the other polynomial
• Combine like terms after multiplication
• Simplify the resulting polynomial

Polynomial Division

Long Division

• Write dividend and divisor in descending order of degree
• Divide leading term of dividend by leading term of divisor
• Multiply divisor by result and subtract from dividend
• Repeat steps until remainder is 0 or has lower degree than divisor
• Quotient is the result of division, and remainder is the left-over term

Synthetic Division

• Use for dividing a polynomial by a linear divisor (x - c)
• Write coefficients of dividend in a table, with leading coefficient on the left
• Bring down the leading coefficient
• Multiply divisor's coefficient (c) by brought-down coefficient, and add to next column
• Repeat step until last column is reached
• Last column represents the quotient and remainder

Factoring

• Express a polynomial as a product of simpler polynomials
• Techniques include:
  • Factoring out greatest common factors (GCFs)
  • Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b))
  • Using the sum and difference formulas (a^2 + 2ab + b^2 = (a + b)^2, a^2 - 2ab + b^2 = (a - b)^2)
  • Factoring by grouping

Learn how to add, subtract, and multiply polynomials by combining like terms and using the distributive property. Practice simplifying polynomials and mastering these essential algebra skills.