Notes on Division Algorithms: Numbers and Polynomials

Division Basics: Long Division (Numbers)

• Goal: Understand divisor, dividend, quotient, and remainder, and see how the division algorithm expresses the dividend in terms of the divisor and quotient plus remainder.
• Terms:
  • Divisor: the number you divide by. In the example, the divisor is 26.
  • Dividend: the number being divided. In the example, the dividend is 45.81.
  • Quotient: the result of the division. In the example, the quotient is 1.76 (ignoring the remainder for the moment).
  • Remainder: what is left over after performing the division, with a value smaller than the divisor. In the example, the remainder is 5.
• Process (the long-division steps): Divide → Multiply → Subtract → Bring down → Repeat.
• Key observations:
  • The remainder is always less than the divisor: in the example, 5 < 26.
  • One can write the division in two equivalent forms:
  • Form 1 (dividend = divisor × quotient + remainder):
    D = d \cdot Q + R
  • Form 2 (quotient plus fractional remainder):
    \frac{D}{d} = Q + \frac{R}{d}
• Numerical example (45.81 ÷ 26):
  • 26 goes into 45 about 1 time. 26 × 1 = 26.
  • Subtract: 45 − 26 = 19.
  • Bring down the next digit (the 8) to form 198.
  • 26 goes into 198 about 7 times. 26 × 7 = 182.
  • Subtract: 198 − 182 = 16.
  • Bring down the next digit (the 1) to form 161.
  • 26 goes into 161 about 6 times. 26 × 6 = 156.
  • Subtract: 161 − 156 = 5.
  • No more digits to bring down; remainder = 5.
  • Therefore, the division yields a quotient of 1.76 with remainder 5, and the relationships hold:
  • 45.81 = 26 \cdot 1.76 + 5
  • \frac{45.81}{26} = 1.76 + \frac{5}{26}
• Practical interpretation:
  • The dividend divided by the divisor equals the quotient plus the remainder over the divisor.
  • If the remainder were 0, the divisor would divide the dividend exactly, so the dividend would factor as the divisor times the quotient, and both the divisor and the quotient would be factors of the dividend.

Division of Polynomials: Setup and Notation

• When dividing polynomials, you use the same algorithm and terminology, but with function notation:
  • Dividend: the polynomial f(x) (the one being divided).
  • Divisor: the polynomial d(x).
  • Quotient: the polynomial q(x).
  • Remainder: the polynomial r(x).
• The division algorithm for polynomials states:
  f(x) = d(x)\,q(x) + r(x)
• If the remainder is zero (r(x) = 0), then the divisor divides the dividend evenly and we have a factorization:
  • f(x) = d(x)\,q(x)
  • In this case, both the divisor and the quotient are factors of f(x).
• Form two (equivalent) ways to express the relationship in polynomial long division:
  • Form 1: f(x) = d(x)\,q(x) + r(x)
  • Form 2 (quotient form): \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}
• The setup and steps mirror numeric long division: divide leading terms, multiply the divisor by the current quotient term, subtract, bring down the next term, and repeat until no terms remain (or until the remainder is of degree less than the divisor).
• Important practical tip:
  • When a term is missing in the dividend (e.g., no x^3 term or no constant term), insert a zero coefficient to keep alignment in the long division steps. If the missing term is truly zero, you do not write it in the final expression, but you must represent it internally for correct bookkeeping.

Example 1: Polynomial long division with remainder 0

• Dividend: f(x) = x^3 - 2x^2 - 5x + 6
• Divisor: d(x) = x - 3
• Goal: compute the quotient q(x) and remainder r(x).
• Steps:
  1) Divide leading terms: \frac{x^3}{x} = x^2. Multiply: x^2\cdot(x-3) = x^3 - 3x^2. Subtract:
  (x^3 - 2x^2) - (x^3 - 3x^2) = x^2.
  2) Bring down the next term (-5x): now we have x^2 - 5x. Divide: \frac{x^2}{x} = x. Multiply: x\cdot(x-3) = x^2 - 3x. Subtract:
  (x^2 - 5x) - (x^2 - 3x) = -2x. Bring down the next term (+6): -2x + 6. Divide: \frac{-2x}{x} = -2. Multiply: -2\cdot(x-3) = -2x + 6. Subtract:
  (-2x + 6) - (-2x + 6) = 0. Remainder = 0.
• Result:
  • Quotient: q(x) = x^2 + x - 2
  • Remainder: r(x) = 0
• Verification (factorization):
  • Since r(x) = 0, f(x) = d(x)\,q(x) = (x-3)(x^2 + x - 2).
  • Further factorization: x^2 + x - 2 = (x+2)(x-1), so f(x) = (x-3)(x+2)(x-1).
• Observations:
  • The division algorithm holds: f(x) = d(x)\,q(x) + r(x) with r(x) = 0.
  • When r(x) = 0, the dividend factors into the divisor times the quotient.

Example 2: Polynomial long division with missing terms and a nonzero remainder

• Dividend: f(x) = 4x^4 + 6x
• Divisor: d(x) = x - 4
• Note: The dividend has missing terms for degrees 3, 2, and the constant term. To perform long division cleanly, introduce zero coefficients for those terms:
  • Represented as: f(x) = 4x^4 + 0x^3 + 0x^2 + 6x + 0
• Steps (as described in the transcript):
  1) Divide leading terms: \frac{4x^4}{x} = 4x^3. Multiply: 4x^3\cdot(x-4) = 4x^4 - 16x^3. Subtract:
  (4x^4 + 0x^3) - (4x^4 - 16x^3) = 16x^3. Bring down the next term (0x^2): we now have 16x^3 + 0x^2 + 6x.
  2) Divide leading terms: \frac{16x^3}{x} = 16x^2. Multiply: 16x^2\cdot(x-4) = 16x^3 - 64x^2. Subtract:
  (16x^3 + 0x^2) - (16x^3 - 64x^2) = 64x^2. Bring down the next term (+6x): 64x^2 + 6x.
  3) Divide leading terms: \frac{64x^2}{x} = 64x. Multiply: 64x\cdot(x-4) = 64x^2 - 256x. Subtract:
  (64x^2 + 6x) - (64x^2 - 256x) = 262x. Bring down the constant term (0): 262x + 0.
  4) Divide leading terms: \frac{262x}{x} = 262. Multiply: 262\cdot(x-4) = 262x - 1048. Subtract:
  (262x + 0) - (262x - 1048) = 1048. Remainder = 1048.
• Result (as stated in the transcript, using the provided numbers):
  • Quotient: q(x) = 4x^3 + 16x^2 + 64x + 262
  • Remainder: r(x) = 1048
• Fraction form of the division:
  • \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)} = 4x^3 + 16x^2 + 64x + 262 + \frac{1048}{x-4}.
• Note on the transcript numbers:
  • The transcript presents a final remainder of 984 in a similar setup, and a quotient of 4x^3 + 16x^2 + 60x + 246 in that specific example. The exact numeric values depend on the precise dividend terms used in that problem. The main idea remains:
  • For any polynomial division, you can express the result as f(x) = d(x)\,q(x) + r(x) with degree constraints on r(x).
  • If r(x) ≠ 0, you may also write \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}.
• Observations:
  • Even when some coefficients are zero (missing terms), long division proceeds by aligning like terms; you insert zeros to keep track of degrees.
  • A nonzero remainder means the divisor does not factor the dividend exactly in that step; only when the remainder is zero do we obtain a clean factorization f(x) = d(x) q(x).

Key takeaways, connections, and implications

• Division algorithm generalizes from numbers to polynomials:
  • For polynomials, the same structural relationship holds: f(x) = d(x)\,q(x) + r(x).
  • If the remainder is zero, the divisor is a factor of the dividend and the quotient is the corresponding cofactor.
• In the context of zeros of polynomials:
  • Long division helps reveal factorization, which in turn yields zeros (roots) by solving d(x) = 0 and/or q(x) = 0 in appropriate cases. The transcript foreshadows that zeros will be the topic in the next part, and that long division is only one method among others.
• Practical notes for doing long division with polynomials:
  • Always align terms by degree; insert zero coefficients for missing terms to keep alignment correct.
  • Write both forms of the result (Form 1 and Form 2) to see the relationship between the quotient and the remainder transparently.
  • If you encounter a remainder of zero, you have factored the dividend; you can factor the quotient further if possible (e.g., factoring x^2 + x - 2 as (x+2)(x-1)).
• Ethical and philosophical note (process mindset):
  • The instructor emphasizes patience and following the algorithm step-by-step, illustrating why procedural rigor matters in mathematics and how structure (division algorithm) underpins later topics like factoring and finding zeros.

Quick reference: key formulas to memorize

• Numeric division:
  D = d \cdot Q + R, \quad \frac{D}{d} = Q + \frac{R}{d},
  where 0 \le R < d.

• Polynomial division (general):
  f(x) = d(x)\,q(x) + r(x),
  \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}.

• Factorization check (remainder zero):

  • If r(x) = 0, then f(x) = d(x)\,q(x) and both d(x) and q(x) are factors of f(x).

Connections to the course flow

• This section builds the bridge from factoring (where possible) to division when factoring is not straightforward.
• It sets up the technique of polynomial long division as a tool to analyze polynomials, with the goal of eventually identifying zeros (roots) via factorization or other methods in future lessons.