Polynomials & Polynomial Equations – Division Techniques

Polynomial Fundamentals

  • A polynomial expression P(x) has the general form
    \displaystyle P(x)=anx^n+a{n-1}x^{n-1}+a{n-2}x^{n-2}+\dots+a1x+a0 • n is a non-negative integer (the degree). • Coefficients ai are real numbers (with a_n\neq 0).
    • Example (mixed order, then re-ordered):
    – Given 3x^2+2x^4-x^3+x^2-2x+1.
    – Re-order → 2x^4-x^3+4x^2-2x+1.
    • Key terms:
    – Leading term = highest power term (e.g.
    3x^5 in 3x^5+\dots).
    – Degree = highest exponent (e.g.
    5).
    – Leading coefficient = coefficient of leading term (e.g.
    3).

Importance of Focus

  • Slide illusion (black dot & shrinking gray haze) used to emphasize:
    • FOCUS and CONCENTRATION are critical disciplines when learning new mathematical skills.
    • Clear attention → clearer understanding of math concepts.

Long Division of Polynomials

  • Purpose: divide any polynomial P(x) by another polynomial D(x) (degree ≥ 1) to obtain
    P(x)=Q(x)\,D(x)+R(x) with \deg R<\deg D.

  • Algorithm (4-step cycle):

    1. Divide the first (leading) term of the dividend by the leading term of the divisor to get the next term of the quotient.

    2. Multiply the entire divisor by this quotient term.

    3. Subtract the product from the current dividend (remainder update).

    4. Bring down the next term of the dividend and repeat until the remainder’s degree is lower than the divisor or is zero.

  • Memory device from transcript: “Divide → Multiply → Subtract → Bring Down → Repeat.”

Worked Long-Division Example (fully shown on slides)

  • Divide (4x^6+13x^5-4x^4+31x^3-2x^2+7x+6) by (x+4).
    • Detailed board work ends with
    4x^6+13x^5-4x^4+31x^3-2x^2+7x+6=(4x^5-3x^4+8x^3-x^2+2x-1)(x+4)+10.

Additional long-division example

  • Divide (3x^3-2x^2+7x+6) by (x+4) → quotient =3x^2-14x+63, remainder =-246.

Guided Long-Division Practice Lists

  • Students asked to write answers in the form P(x)=Q(x)D(x)+R(x). • Set 1:

    1. (x^3+2x^2-x-2)\div(x-1)

    2. (x^5+2x^4+6x+4x^2+9x^3-2)\div(x+2)

    3. (x^3+7x^2+5x-25)\div(x+5)

    4. (2x^3-13x^2-5x+100)\div(x-5).
      • Set 2 (later slide):

    5. (x^3+x^2-22x-25)\div(x+2)

    6. (4x^4+21x^3-26x^2+28x-10)\div(x+5)

    7. (6x^3-25x^2-31x+20)\div(3x-2).

Simplified Long Division (Coefficient-Only Method)

  • Works like classical long division but manipulates only the numerical coefficients; variables are re-inserted at end.

  • Guidelines:

    1. Arrange divisor & dividend in descending order.

    2. Work strictly with coefficient rows.

    3. Apply the same divide-multiply-subtract cycle.

    4. Repeat until remainder degree < divisor degree.

    5. Insert proper variables into quotient terms afterward.

Simplified Example 1

  • Divide (4x^5+6x^4+5x^2-x-10) by (2x^2+3).
    • Coefficient setup table produces quotient 2x^3+3x^2-3x-2 with remainder 8x-4.
    • Final statement:
    4x^5+6x^4+5x^2-x-10=(2x^3+3x^2-3x-2)(2x^2+3)+(8x-4).

Simplified Example 2

  • Divide (4x^5-25x^4+40x^3+3x^2-18x) by (x^2-6x+9)
    • Coefficient table yields quotient 4x^3-x^2-2x (remainder 0 → exact division).
    • Interpretation: x^2-6x+9=(x-3)^2 so original polynomial is divisible by (x-3)^2.

Student Practice (Simplified Long Division)

  • Same three problems as long-division Set 2 (see above) are to be re-done via simplified method.

Synthetic Division

  • Applicable only when divisor is a linear binomial of the form (x-c) or equivalent k x - c convertible to x-c/k by factoring.

  • Advantages: fastest; uses only coefficients.

Standard Synthetic Steps (for x-c)

  1. Write the zero of the divisor (i.e. c from x-c) at left.

  2. List coefficients of the dividend in descending powers; insert zeros for missing degrees.

  3. Bring down the leading coefficient.

  4. Multiply the brought-down value by c, write under next coefficient, add, repeat.

  5. Final row = coefficients of quotient; last value = remainder.

Example (A) – divisor x+2

  • Divide 8x^3+10x^2+7x+6 by (x+2) (so c=-2).
    • Synthetic tableau gives quotient 8x^2-6x+19 with remainder -32.
    • Result:
    8x^3+10x^2+7x+6=(x+2)(8x^2-6x+19)-32.

Example (B) – divisor 3x-2

  • Must rewrite: (3x-2)=3\bigl(x-\tfrac{2}{3}\bigr).

  • Two equivalent synthetic approaches shown on slide:
    a) Treat divisor as x-\tfrac{2}{3} ⇒ use c=\tfrac{2}{3}, then divide remainder by leading factor 3 afterwards.
    b) “Scaled” synthetic table: factor 3 tracked inside.

  • Provided answer: Quotient =2x^2-4x+3, remainder \dfrac{6}{3}=2 when final divisor expressed as 3x-2.

Worked Practice

  • Five exercises listed; two fully solved in slides:

    1. 5x^2-17x-15\div(x-4) → quotient 5x+3, remainder -3.

    2. 2x^4+x^3-19x^2+18x+5\div(2x-5) → quotient 2x^3+6x^2-4x+8, remainder 25.

  • Additional unsolved practice:

    1. 3x^3+x^2-22x-25\div(x-2).

    2. x^3+4x^2-x-25\div(x+5).

    3. 6x^3-5x^2+4x-1\div(3x-1).

  • Second batch (unsolved):

    1. 11x^2+4x^3+8\div(x+3).

    2. 4x^4-2x^3-10x^2+7x+8\div(2x+3).

    3. 4x^3-5x^2+2x-1\div(x-1).

    4. 5x^4+8x^3+x^2+9x+3\div(x+2).

    5. x^4+4x^3+4x^2+2x-3\div(x+3).

Organizing Polynomials in Descending Order

  • Exercise slide: Students must reorder each expression then state degree.

    1. x^5+x^3+x^2-22x+2\,(\deg=5)

    2. 21x^5+5x^4-26x^3+4x^2+28x-10\,(\deg=5)

    3. -2x^4+3x^3-31x+6\,(\deg=4)

    4. 5x^4+x^3+7x^2-25x\,(\deg=4)

    5. 5x^5+x^3+7x^2-25x+5\,(\deg=5).

Summary of Division Techniques

  • Long Division: universally applicable; mirrors numeric long division; must align like terms.

  • Simplified Long Division: coefficient-only shortcut; re-insert variables.

  • Synthetic Division: fastest but limited to linear divisors of form (x-c) (or scalable variant).

  • All methods ultimately satisfy P(x)=Q(x)D(x)+R(x), giving insight into factors, zeros, and remainders.

Conceptual & Practical Significance

  • Division algorithms underpin advanced topics:
    • Factor & Remainder Theorems (e.g.
    R=P(c) in synthetic division).
    • Finding polynomial zeros & multiplicities.
    • Partial-fraction decomposition.
    • Simplifying rational expressions.

  • Ethical study habit reminder: Maintaining strong focus (illustrated by the gray-haze illusion) leads to deeper comprehension and academic integrity.