Study Notes on Factors, Remainders, and Polynomial Division

Understanding Factors

• Definition of a Factor

  • A factor is defined as an expression or number that divides another expression or number evenly without a remainder.

  • Example: Identifying factors of 10.

    • Positive pairs: 1 × 10, 2 × 5.

    • Negative pairs: -1 × -10, -2 × -5.

  • Both positive and negative factors must be considered.

  • Remainder Principle: The remainder when dividing a number by one of its factors must equal zero.

Examples of Factors

• Example with Division

  • When dividing 10 by 2 (a factor), the quotient is 5 and the remainder is 0.

  • Conclusion: If the divisor is a factor of the dividend, the remainder is zero. Conversely, if the remainder is zero, the divisor is a factor of the dividend.

Extending the Concept to Polynomials

• Polynomial Example

  • Dividing polynomial: x^2 - 5x + 6 by x - 2 using synthetic division.

  • Steps for Synthetic Division:

    1. Draw half box symbol.

    2. Write 2 (value of a from x - a) to the left of the box.

    3. Bring down the coefficients of the polynomial (1 for x^2, -5 for x, 6 for the constant).

    4. Multiply the first coefficient (1) by 2 = 2, write under -5.

    5. Add to get -3.

    6. Multiply -3 by 2 = -6, write under 6.

    7. Add to get 0.

  • Interpretation:

    • The last number represents the remainder. Here, the remainder is 0, concluding that x - 2 is a factor of x^2 - 5x + 6.

Evaluating Polynomials and the Remainder Theorem

• Remainder Theorem

  • States that the remainder from the division of a polynomial p(x) by a divisor d(x) in the form x - a equals p(a).

  • In the current example, evaluating p(2) leads to:

    • 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0.

  • Consequently, this reaffirms that the remainder from the synthetic division aligns with the remainder theorem's prediction concerning factors.

Factoring the Polynomial

• Factorization Example

  • Attempting to factor the polynomial x^2 - 5x + 6 confirms the factor.

  • Factorization result: x - 2 and x - 3 are valid factors.

  • Validation by substitution: Evaluating at x = 3 yields:

    • 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0.

Conclusion: The Factor Theorem

• Statement of the Factor Theorem

  • If x - a is a factor of polynomial p(x), then p(a) = 0, meaning a is a root of the polynomial.

  • Conversely, if p(a) = 0, then x - a is a factor of the polynomial p(x).

  • The factor theorem expands our ability to find factors of polynomials by using known roots.

Future Directions

• Next Video Preview

  • Upcoming discussion will cover methods to fully factor polynomials, including those of degree three or higher.

  • The relevance of the factor and remainder theorems will be further explored in applications.