math 5.4

Polynomial and Factoring Basics

  • Given expression: 5t + (5t - 6) + 4t²

    • Importance of correct order: Write in descending order.

    • Standard form rearrangement: 4t² + 5t - 6

Factoring Polynomials

  • Goal of factoring: Identifying factors of negative six.

Factors of Negative Six

  • Three pairs of negative six:

    • 1 x (-6)

    • -1 x 6

    • 2 x (-3)

    • -2 x 3

Checking Factors

  • Method involves multiplying the terms:

    • E.g., Checking (4t and -3) vs (t and 2)

    • Calculation:

    • 4t x (-3) = -12t

    • t x 2 = 2t

    • Adjusting coefficients to achieve 5t.

    • Combination yields: 4t - 3 and t + 2 = 0.

Special Factoring Techniques

Difference of Squares

  • Definition: The difference of squares requires both terms to be perfect squares:

    • Example formula: x² - y² = (x - y)(x + y)

    • Sign difference: One term is positive, the other is negative.

  • Example: t² - 81

    • Factorization: (t - 9)(t + 9)

Prime Polynomials

  • A polynomial is prime if it cannot be factored further.

    • Example: 10 not a perfect square.

Conditions for Perfect Square Trinomials

  • For a trinomial to be a perfect square, two conditions must be true:

    • Each of the first and last terms must be perfect squares.

    • The middle term must equal twice the product of the square roots of the first and last terms.

    • Example:

    • x² + 8x + 16 = (x + 4)²

Writing Key Formulas for Perfect Square Trinomials

  • (x + y)² = x² + 2xy + y²

  • (x - y)² = x² - 2xy + y²

Example Exercises Note

  • Start with: k² + 20k + 100

    • Apply perfect square factorization: (k + 10)²

  • Further exercises for practice include:

    • x² + 12x + 36 = (x + 6)²

Prime vs Composite Review

  • A polynomial is considered prime if factors cannot be derived.

    • Example: 6a + 5 does not adhere to conditions for factorization.

Final Notes

  • Calculation methods for checking factors involve backward multiplication.

  • Remember distinction between perfect squares vs factors.

  • Concept of diligence in mathematical reasoning discussed and encouraged.

Difference of Cubes

  • Formula: x³ - y³ = (x - y)(x² + xy + y²)

    • Difference of cubes setup with variables captured:

    • 125p³ - 8 = Factor by identifying x = 5p and y = 2

    • Resultant format: (5p - 2)(25p² + 10p + 4)

Sum of Cubes

  • Formula: x³ + y³ = (x + y)(x² - xy + y²)

    • Example: a³ + 1000b⁶ = (a + 10b²)(a² - 10ab² + 100)

Overall Conclusions

  • Encourage students to practice consistently with pairs, as many polynomials can be prime, and many require careful analysis on factors and squares.

  • Keep engaging with the material by reinforcing the use of definitions, formulas, and numeric examples in various forms of polynomials and trinomial with squares.

  • Reinforce understanding that primes cannot be factored beyond their base components.

  • Always check your solutions through direct multiplication and verification for accuracy.