Factoring Polynomials and Division

Overview

• Lecture covers factoring polynomials, polynomial division, identifying zeros, end behavior, and solving polynomial equations.
• Emphasis on rewriting polynomials as products of linear factors and checking remainders for division errors.
• Examples illustrate box (grid) method and long division, factoring quadratics, and graph interpretation.

Factoring Using Division (Example 1)

• Given polynomial B(x) and known factor (x - 4), divide to get remaining factor.
• Use box/grid method dividing by a binomial; fill terms by reverse multiplication.
• Division steps (summary):
  • First quotient term x^2 because x^2 * x = x^3.
  • Multiply: x^2 * (-4) = -4x^2.
  • Cancel x^2 terms by adding +4x^2 in next quotient term.
  • Next quotient term 4x gives 4x * (-4) = -16x.
  • To reach -21x, next quotient term is -5 (since -16x + -5x = -21x).
  • Constant from -5 * -4 = +20 matches constant term → remainder 0.
• Result after division: B(x) = (x - 4)(x^2 + 4x - 5).
• Further factor quadratic x^2 + 4x - 5:
  • Need factors of -5 that add to +4 → +5 and -1.
  • Quadratic factors as (x + 5)(x - 1).
• Final linear-factorization: B(x) = (x - 4)(x + 5)(x - 1).

Division Remainder Check (Example 2)

• If given that (x + 9) is a factor, remainder must equal 0 when dividing.
• Observation: computed remainder = 990 → indicates division error.
• Rule: factor claim implies remainder = 0; nonzero remainder proves a mistake.

Correcting Textbook Error and Successive Factoring (Example 3)

• Textbook misstated factor: correct factor is (x - 5), not (x + 5).
• Divide polynomial by (x - 5) using grid/box method:
  • Leading quotient term x^3 * x gives x^4, etc.; follow reverse multiplication.
  • Successive quotient terms found: x^3, +3x^2, -6x^2 etc. (see division logic).
  • After division obtain cubic, then test zeroes from linear factors.
• Use zeros corresponding to factors:
  • For factor (x - 5), zero is x = 5.
  • For factor (x + 1), zero is x = -1; plug into reduced polynomial to confirm zero.
  • Plugging -1 into cubic expression yields 0, so (x + 1) is a factor.
• Continue division by (x + 1) to reduce polynomial further, then factor final quadratic:
  • Final quadratic x^2 + 2x - 8 factors to (x - 2)(x + 4).
• Final linear factors (from example): (x - 5)(x + 1)(x - 2)(x + 4) (as derived from steps).*

Multiplying End Constants (Matching Constants) (Example 4)

• Constant term of polynomial equals product of constant terms of linear factors.
• Example calculations:
  • (-2)(-3)(7) = 42 → matches case A.
  • (2)(-3)(7) = -42 → matches case B.
  • (-2)(-3)(7)(1/2) = 21 → matches case C.
  • (-2)(-3)(7)(5) = 210 → matches case D or E context (sign matters).
  • (-2)(-3)(7)(-5) = -210 → matches remaining case.
• Procedure: multiply constant factors, include any fractional multipliers, check sign.

Solving a Polynomial Equation By Factoring (Example 5)

• Solve equation of form (x - 2)(x - 4) = 8 (example).
• Expand left side: x*x = x^2; cross terms -4x -2x = -6x; constant (-2)(-4) = +8.
• Equation becomes x^2 - 6x + 8 = 8.
• Subtract 8 both sides → x^2 - 6x = 0.
• Factor common x: x(x - 6) = 0 → solutions x = 0 and x = 6.
• Principle: if product equals zero, at least one factor equals zero.*

End Behavior From Graph (Example 6)

• Given graph: as x → +∞, f(x) → +∞; as x → -∞, f(x) → -∞.
• This corresponds to a polynomial with odd degree and positive leading coefficient.
• Correct statement: right end up, left end down.

Factoring And Sketching From Known Factor (Example 7)

• Given P(x) with known factor (x + 4); divide to find quotient:

  • Division yields x^2 - x - 2 after dividing x^3 + 3x^2 - 6x - 8 by (x + 4).
  • Confirm remainder 0.

• Factor quotient quadratic x^2 - x - 2:

  • Factors of -2 summing to -1 → -2 and +1.
  • Quadratic factors as (x - 2)(x + 1).

• Full linear factorization: P(x) = (x + 4)(x - 2)(x + 1).

• Zeros and intercepts:

  • Zeros at x = -4, x = 2, x = -1.
  • y-intercept equals constant term -8.

• Degree and leading coefficient:

  • Degree 3 (odd), leading coefficient positive → end behavior down on left, up on right.

• Sketch essentials:

  • Start down on left, cross x-axis at -4, then -1, then 2, end up on right.
  • y-intercept at (0, -8).

Key Terms And Definitions

• Factor / Factorization: express polynomial as product of polynomials of lower degree.
• Linear Factor: degree-one factor of form (x - a) or (x + b).
• Zero/Root: value x0 where polynomial evaluates to zero; related to linear factor (x - x0).
• Remainder Theorem: remainder of polynomial division by (x - a) equals f(a).
• End Behavior: how f(x) behaves as x → ±∞; determined by degree (even/odd) and sign of leading coefficient.

Action Items / Next Steps

• When given a claimed factor, always check remainder = 0 by substitution or division.
• Use box/grid method or long division consistently; choose method by convenience.
• For quadratics with a = 1, factor by finding integer pairs multiplying to constant term and summing to middle coefficient.
• For sketching:
  • Identify zeros, multiplicities, y-intercept, degree, and leading coefficient sign.
  • Use end behavior to place arms of graph correctly.