Lesson 42: Factoring and the Remainder Theorem
Factoring is a key method for solving polynomial equations and analyzing functions. The Remainder Theorem provides a quick way to evaluate polynomials and determine factors.
Factoring Polynomials
Common factoring techniques:
- Greatest Common Factor (GCF): Factor out the largest common term.
- Factoring by grouping: Group terms and factor each group.
- Difference of squares: a² − b² = (a − b)(a + b)
- Sum/Difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
- Trinomials: ax² + bx + c → factor into two binomials
Example: Factor 2x³ + 4x² − 6x
- GCF = 2x → 2x(x² + 2x − 3)
- Factor trinomial: x² + 2x − 3 = (x + 3)(x − 1)
- Final factorization: 2x(x + 3)(x − 1)
The Remainder Theorem
The Remainder Theorem states:
If a polynomial f(x) is divided by (x − c), the remainder is f(c).
Example: f(x) = 2x³ − 3x² + x − 5, divide by (x − 2)
- Remainder = f(2) = 2(2³) − 3(2²) + 2 − 5 = 16 − 12 + 2 − 5 = 1
Implication: If f(c) = 0, then (x − c) is a factor of f(x).
Practice Problems
- Factor: x³ + 3x² − 4x − 12
- Factor: 2x³ − 8x² + 6x
- Use Remainder Theorem: f(x) = x³ − 6x² + 11x − 6, divide by (x − 1)
- Use Remainder Theorem: f(x) = 3x³ + x² − 5x + 2, divide by (x + 2)
- Determine if (x − 2) is a factor of f(x) = x³ − 4x² + x + 6