Comprehensive Study Notes: Factor Theorem, Sequences, and Polynomial Division

Factor Theorem and Polynomial Factorization

• Factor Theorem concept: If a polynomial P(x) evaluated at c equals zero, P(c) = 0, then (x - c) is a factor of P(x).
• Useful for factoring polynomials and finding roots.
• Classic factorization examples from transcript:
  • 2x^2 - x - 6 = (2x + 3)(x - 2)
  • x^2 - 9 = (x + 3)(x - 3)
  • 4x^2 + 20x + 25 = (2x + 5)^2
  • x^3 + 4x^2 + 3x = x(x^2 + 4x + 3) = x(x + 1)(x + 3)
• Summary of factorization strategy:
  • Factor quadratics into products of linear terms when possible.
  • Factor out common factors, then factor the remaining quadratic or higher-degree polynomial.
  • Use the Factor Theorem to test potential roots, then factor accordingly.

Geometric Sequences and Sums

• Geometric sequence (Geometric Progression): a1, a2, a3, … where each term is obtained by multiplying the previous term by a nonzero constant r (the common ratio).
• Key definitions:
  • General term: $a<em>n = a</em>1 r^{\,n-1}$
  • Sum of first n terms (geometric series): $S<em>n = \frac{a</em>1\,(1 - r^n)}{1 - r}, \quad r \neq 1$
  • If |r| < 1, the infinite sum converges as n → ∞.
• Examples from transcript:
  • (a) 4, 12, 36, 108, 324, … → a1 = 4, r = 3
  • (b) 3, -6, 12, -24, 48, -96, … → a1 = 3, r = -2
  • (c) -250, 50, -10, 2, … → a1 = -250, r = -\tfrac{1}{5} = -0.2
• How to identify r:
  • r is the quotient of any term by its previous term (for n ≥ 2): $r = \dfrac{a<em>{n}}{a</em>{n-1}}$
• Note on finite vs infinite sums:
  • Geometric sums with |r| < 1 converge to a finite value as n → ∞; those with |r| ≥ 1 diverge.

Sequences: Finite vs Infinite; Terms and Patterns

• A sequence is an ordered list of elements (often numbers) that follows a rule or pattern.
• Notation:
  • The first term is a1, the second term a2, and the nth term is an.
• Finite sequence:
  • Has a first term and a last term (ellipsis indicates continuation until that last term).
  • Examples:
  • 5, 10, 15, 20, 25
  • 15, 30, 45, 60, 75
• Infinite sequence:
  • No last term; continues indefinitely.
  • Examples:
  • 1, 2, 3, 4, 5, …
  • 7, 14, 21, 28, 35, …
• Pattern in a sequence:
  • Odd numbers example: 1, 3, 5, 7, 9, 11, …
  • Even numbers example: 2, 4, 6, 8, 10, 12, …
• Ellipsis (…): Indicates continuation according to the rule of the sequence.
• Three common sequence types shown in transcript tasks:
  • Identify finite vs infinite using ellipsis presence and given first/last terms.
  • Recognize simple patterns such as adding a constant (arithmetic) or multiplying by a constant (geometric).

Arithmetic Sequences: Definition and Sums

• Arithmetic sequence: consecutive terms differ by a constant difference d (the common difference).
• Key formulas:
  • nth term: $a<em>n = a</em>1 + (n - 1) d$
  • Sum of the first n terms: $S<em>n = \frac{n}{2}\left(a</em>1 + a<em>n\right) = \frac{n}{2}\left[2a</em>1 + (n - 1) d\right]$
• Worked examples from transcript:
  • Example 1: 5, 9, 13, 17, … with a1 = 5, d = 4
  • a10 = 5 + (10 - 1)\cdot 4 = 41
  • S10 = \frac{10}{2} (5 + 41) = 230
  • Example 2: -2, -5, -8, -11, … with a1 = -2, d = -3
  • a20 = -2 + 19(-3) = -59
  • S20 = \frac{20}{2} (-2 + -59) = -610
• Additional notes:
  • For any arithmetic sequence, d can be found from two known terms: d = (an - a1) / (n - 1).
  • Example solving for d: a1 = 13, a_n = 67, n = 7 ⇒ d = (67 - 13) / (7 - 1) = 9.

Arithmetic Means and Missing Terms

• Arithmetic means: used to fill in missing terms when a1, an, and n are known.
• Key relation: $a<em>n = a</em>1 + (n - 1) d\quad\Rightarrow\quad d = \frac{a<em>n - a</em>1}{n - 1}$
• Example calculations from transcript:
  • If a1 = 13, an = 67, and n = 7, then $d = \frac{67 - 13}{7 - 1} = 9.$
  • If a1 = 0.3, and a_n = 3.5 with n - 1 = 8, then $d = \frac{3.5 - 0.3}{8} = 0.4.$
• Transpose/solving steps were noted for solving these equations (e.g., isolating d).

Remainder Theorem and Factor Theorem

• Remainder Theorem: Given a polynomial P(x), dividing by (x - c) yields remainder P(c).
• Factor Theorem: If P(c) = 0, then (x - c) is a factor of P(x).
• Examples (from transcript):
  • P(x) = 6x^3 - 13x^2 + 16x - 15; evaluate at x = -2:
  • P(-2) = 6(-2)^3 - 13(-2)^2 + 16(-2) - 15 = -192 - 52 - 32 - 15 = -291
  • Therefore remainder when dividing by (x + 2) is -291 (illustrating the remainder concept).
  • P(x) = x^3 + 6x^2 + x - 12; x = -4:
  • P(-4) = (-4)^3 + 6(-4)^2 + (-4) - 12 = -64 + 96 - 4 - 12 = 16
  • So remainder is 16 when dividing by (x + 4).
  • P(x) = 3x^3 - 16x^2 + 19x - 12; x = 1:
  • P(1) = 3 - 16 + 19 - 12 = -6
• Long division of polynomials (examples):
  • Dividing by linear binomials (x - c) or by linear polynomials such as (2x - 3).
  • Example 1 (long division): P(x) = 6x^3 - 13x^2 + 16x - 15 by (2x - 3)
  • Quotient: $3x^2 - 2x + 5\,;\; \text{Remainder } 0.$
  • Example 2 (synthetic division for (x + 2)): P(x) = 6x^3 - 13x^2 + 16x - 15 divided by (x + 2)
  • Coefficients: 6, -13, 16, -15; use root c = -2
  • Bring down 6; 6(-2) = -12; sum -13 + (-12) = -25; (-25)(-2) = 50; 16 + 50 = 66; 66(-2) = -132; -15 + (-132) = -147
  • Quotient: $6x^2 - 25x + 66;\; \text{Remainder } -147.$
  • Example 3 (synthetic division for (x - 3)): P(x) = 3x^3 - 11x^2 + 12x - 18 divided by (x - 3)
  • Coefficients: 3, -11, 12, -18; root c = 3
  • Bring down 3; 3(3) = 9; -11 + 9 = -2; (-2)(3) = -6; 12 + (-6) = 6; (6)(3) = 18; -18 + 18 = 0
  • Quotient: $3x^2 - 2x + 6;\; \text{Remainder } 0.$
  • Example 4 (synthetic division for (x - 5)): P(x) = 4x^3 - 26x^2 + 32x - 10 divided by (x - 5)
  • Coefficients: 4, -26, 32, -10; root c = 5
  • Bring down 4; 4(5) = 20; -26 + 20 = -6; (-6)(5) = -30; 32 + (-30) = 2; 2(5) = 10; -10 + 10 = 0
  • Quotient: $4x^2 - 6x + 2;\; \text{Remainder } 0.$

Division Methods: Long Division vs Synthetic Division

• Synthetic division is a shortcut for dividing by a binomial of the form (x - c).
• Steps for synthetic division (example with P(x) = 6x^3 - 13x^2 + 16x - 15 and divisor x + 2, i.e., c = -2):
  • Write coefficients: 6, -13, 16, -15
  • Bring down the leading coefficient: 6
  • Multiply by c: 6(-2) = -12; add to next coefficient: -13 + (-12) = -25
  • Multiply by c: -25(-2) = 50; add to next: 16 + 50 = 66
  • Multiply by c: 66(-2) = -132; add to last: -15 + (-132) = -147
  • Quotient: $6x^2 - 25x + 66;\; \text{Remainder } -147.$
• Long division example (dividing by 2x - 3):
  • Leading term: (6x^3)/(2x) = 3x^2
  • Multiply and subtract sequentially to obtain quotient and remainder; in this case remainder is 0 and quotient is $3x^2 - 2x + 5.$
• Practical takeaway:
  • Use synthetic division for divisors of the form (x - c).
  • Use long division for divisors of the form (ax + b) with a ≠ 1 or when dividing by quadratics, etc.

Quick Reference: Core Formulas

• Geometric sequence:
  • General term: $a<em>n = a</em>1 r^{n-1}$
  • Sum: $S<em>n = \frac{a</em>1(1 - r^n)}{1 - r}, \quad r \neq 1$
• Arithmetic sequence:
  • nth term: $a<em>n = a</em>1 + (n - 1)d$
  • Sum: $S<em>n = \frac{n}{2}\left(a</em>1 + a<em>n\right) = \frac{n}{2}\left[2a</em>1 + (n - 1)d\right]$
• Remainder Theorem: remainder of P(x) ÷ (x - c) is P(c).
• Factor Theorem: P(c) = 0 ⇔ (x - c) is a factor of P(x).
• Key division tools: synthetic division for (x - c); long division for (ax + b).

Notes on Problem-Solving Flow (from transcript cues)

• Identify whether a sequence is arithmetic or geometric by checking constant difference or constant ratio between consecutive terms.
• For arithmetic sequences, use a1, an, n to find d and then compute sums or specific terms.
• For geometric sequences, use a1 and r to compute terms or sums, and use S_n as needed.
• When factoring polynomials, start with obvious rational roots via the Factor Theorem, then factor the remaining polynomial.
• Use Remainder Theorem to verify potential roots before attempting full factorization.