Notes on Factoring Polynomials of Degree 3

7.1 Factoring Polynomials of Degree 3

Introduction to Factoring Cubics

• Prior Knowledge
    - Students have previously explored methods to factor quadratic (degree 2) polynomial functions.
    - They learned that the quadratic formula can be used to factor quadratic equations effectively, particularly for finding the roots of polynomial functions.
• Challenges with Higher Degree Polynomials
    - Polynomial functions of higher degree (like degree 3 or degree 4) are often more complex to factor.
    - A polynomial is defined as a monomial or a sum of monomials.
    - A monomial consists of a term which can be a constant, a variable, or a product of a constant and one or more variables.
• Historical Context
    - Mathematicians such as Cardano and Ferrari developed methods for factoring certain cubic (degree 3) and quartic (degree 4) polynomials.
    - These historical methods can be cumbersome, limited, and are not widely used.

Special Factorization Forms for Cubics

• Standard Factorization Templates:
    1. For the sum of cubes:
       - $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$
    2. For the difference of cubes:
       - $a^3 - 3a^2b + 3ab^2 - b^3 = (a - b)^3$
    3. The sum of two cubes:
       - $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
    4. The difference of two cubes:
       - $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
    5. General rule for higher degree polynomials:
       - Always consider factoring out the greatest common factors from groups of the given terms.
       Example:
       - $10x^3 - 15x^2 = 5x^2(2x - 3)$

Exercises for Factoring Cubics

• Exercise 7.1.1: Factoring Cubics
    - Instructions: Use factoring by grouping for the following polynomials:
    1. For the polynomial $x^3 + 6x^2 - 5x - 30$:
       - Grouping yields: $(x^3 + 6x^2) - (5x + 30)$
       - Factored result: $-5(x + 6)$
    2. For the polynomial $12x^3 + 2x^2 - 30x - 5$:
       - Grouping leads to: $(12x^3 + 2x^2) - (30x + 5)$
       - Resulting factorization: $2x^2(6x + 1) - 5(6x + 1) = (6x + 1)(2x^2 - 5)$
    3. For the polynomial $8x^3 - 64x^2 + x - 8$:
       - Grouping gives: $8x^3 - 64x^2 + (x - 8)$
       - Final output: $(x - 8)(8x^2 + 1)$

Additional Problems to Factor

• For problems 4-13, use previously learned rules to factor the following cubics:
    4. For $x^3 - 8y^3$:
       - Substitute: $a = x, b^3 = 8y^3$
       - Factorization leads to: $(x - 2y)(x^2 + 2xy + 4y^2)$
    5. For $x^3 + 8$:
       - Substitute: $a = x, b^3 = 8$
       - Factorization yields: $(x + 2)(x^2 - 2x + 4)$
    6. For $x^3 + 1$:
       - Substitute: $a = x, b = 1$
       - Factorization results in: $(x + 1)(x^2 - x + 1)$
    7. For $x^3 + 27$:
       - Substitute: $a = x, b^3 = 27$
       - Provides factorization of: $(x + 3)(x^2 - 3x + 9)$
    8. For $x^3 - 64$:
       - Substitute: $a = x, b^3 = 64$
       - Yielding: $(x - 4)(x^2 + 4x + 16)$
    9. For $x^3 - 6x^2 + 9x$:
       - Direct factorization by taking out common terms: $x(x^2 - 6x + 9)$
       - Further factorization gives: $x(x - 3)^2$
    10. For $2x^3 - 128$:
       - Factor out common factors: $2(x^3 - 64)$
       - Final factorization results in: $2(x - 4)(x^2 + 4x + 16)$
    11. For $64x^3 + 8$:
       - Factor out common factor: $8(8x^3 + 1)$
       - Recognizes also as cubes leading to: $(2x + 1)(4x^2 - 2x + 1)$
    12. For $x^3 - x^2$:
       - Factoring yields: $x^2(x - 1)$
    13. For $6x^3 - x^2$:
       - Final factorization results in: $$ x^2(6x - 1)

Conclusion

• This section focused on factoring polynomials of degree 3, highlighting essential templates, methods, and practice problems to develop proficiency in polynomial factorization. Understanding these forms and approaches is critical for advanced studies in algebra and calculus.