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.