Polynomial Division: Monomial and Long Division Techniques

Defining Polynomials

• Polynomials can be simple numbers or terms multiplied by variables.
• Terms within a polynomial are separated by addition or subtraction.
• Monomial: A polynomial with a single term (e.g., $2x$).
• Binomial: A polynomial with two terms.
• Trinomial: A polynomial with three terms (e.g., $8x^3 + 6x^2 + 4$).
• While explicit definitions may not be tested, recognizing these forms is important for identifying when you are working with polynomials.

Dividing by a Monomial

• When dividing a polynomial by a monomial (a single term with no addition or subtraction), the process involves treating it like a fraction.
• Step 1: Rewrite as a fraction. For example, to divide $(8x^3 + 6x^2 + 4)$ by $2x$, rewrite it as $\frac{8x^3 + 6x^2 + 4}{2x}$ or $(8x^3 + 6x^2 + 4) \div (2x)$.
• Step 2: Divide each term of the numerator by the denominator. This means distributing the division to every term individually:
  $\frac{8x^3}{2x} + \frac{6x^2}{2x} + \frac{4}{2x}$
• Example: To divide $(20y^2 + 12y - 1)$ by $-4y$, you would set it up as:
  $\frac{20y^2}{-4y} + \frac{12y}{-4y} - \frac{1}{-4y}$

Long Division: Numerical Example (Foundation for Polynomial Long Division)

• To understand polynomial long division, it helps to recall traditional long division with numbers.
• **Example: ** Divide $875$ by $25$.
  1. Set up: $25 \text{ | } 875$ .
  2. Estimate the first digit: Determine how many times $25$ goes into $87$. Three times ($3 \times 25 = 75$) is the closest without going over. Four times ($4 \times 25 = 100$) would be too much.
  3. Multiply and Subtract: Write $3$ above the $7$ in $875$. Multiply $3 \times 25 = 75$. Subtract $75$ from $87$ to get $12$.
  4. Bring Down: Bring down the next digit (the $5$) to form $125$.
  5. Repeat: Determine how many times $25$ goes into $125$. This is exactly five times ($5 \times 25 = 125$).
  6. Multiply and Subtract: Write $5$ next to $3$ above the $5$ in $875$. Multiply $5 \times 25 = 125$. Subtract $125$ from $125$ to get $0$.
  7. The quotient is $35$.

Polynomial Long Division: The Process

• The methodology for polynomial long division mirrors the numerical process.
• Step 1: Focus on Leading Terms. Only consider the first term of the divisor and the first term of the dividend.
• Example Scenario: Dividing a polynomial (starting with $2x^2 + 7x$) by $(x+3)$.
  • Ask: "What times $x$ (first term of divisor) would give us $2x^2$ (first term of dividend)?" The answer is $2x$.
• Step 2: Place Term in Quotient and Multiply. Write $2x$ above the dividend, aligning it with the $x$ term. Then, multiply this $2x$ by the entire divisor $(x+3)$. This yields $2x \times (x+3) = 2x^2 + 6x$.
• Step 3: Subtract the Entire Expression (Critical Step). You must subtract the entire result from the previous step. This means changing the sign of each term in the expression you are subtracting.
  • Subtract $(2x^2 + 6x)$ from $(2x^2 + 7x)$. This is equivalent to $(2x^2 + 7x) - (2x^2 + 6x)$.
  • The first terms should always cancel out: $2x^2 - 2x^2 = 0$.
  • Subtract the next terms: $7x - 6x = x$.
• Step 4: Bring Down and Repeat. Bring down the next term from the original dividend (if any). Then, repeat the process with the new leading term ($x$ in this example) and the divisor $(x+3)$.
  • Here, ask: "What times $x$ would give $x$"? The answer is $1$.
  • Multiply $1$ by $(x+3)$ to get $x+3$.
  • Subtract $(x+3)$ from the current polynomial with the brought-down term (if it was for instance $x+3$). If it was $(x+3)-(x+3)$ it will result in $0$. If the example was $(2x^2 + x - 15) ext{ divided by } (x+3)$, after the first step and bringing down $-15$ you would have $-5x-15$. Then you would ask "What times $x$ is $-5x$?" Answer is $-5$. You would multiply $-5(x+3) = -5x-15$. Then subtract $-5x-15 - (-5x-15)$ to get $0$.

Placeholders for Missing Terms

• Importance: To maintain organization and proper alignment of terms (e.g., $x^2$ terms, $x$ terms, constant terms), it is crucial to include placeholders for any missing terms in the dividend.
• Method: If a polynomial is written in descending order of degree but a term is absent (e.g., an $x$ term in a cubic polynomial), represent it with a zero coefficient.
• Example: If you have $5x^3 - 8x^2 + 7$, the $x$ term (which would be $x^1$) is missing.
• Rewrite it as $5x^3 - 8x^2 + 0x + 7$ before performing long division. This ensures all power positions ($x^3$, $x^2$, $x^1$, $x^0$/constant) are accounted for, allowing for neat vertical alignment during subtraction.

Handling Remainders in Polynomial Long Division

• If, after the division process, you are left with a non-zero polynomial term that cannot be further divided by the leading term of the divisor, this is your remainder.
• Format: The remainder is typically expressed as a fraction, with the remainder as the numerator and the original divisor as the denominator.
• Example: If a remainder of $4$ is obtained and the divisor was $(x+6)$, the remainder would be written as $+\frac{4}{x+6}$ or $\frac{4}{x+6}$ if it was negative, $\frac{-4}{x+6}$ or simply $-\frac{4}{x+6}$ .

Academic Logistics and Grading

• Extra Credit: Extra credit opportunities are often separate from the main exam grades and contribute to the overall score in a distinct category.
• Grading Breakdown: Exam grades ($21\%$ for each of Exams 1, 2, and 3) constitute a significant portion of the total course grade. Homework and labs may initially appear to count for more but adjust as exam scores are entered.
• Checking Grades: Students are encouraged to carefully review their grades and calculations. If a grading error benefits the student, they should generally not report it. However, if an error negatively impacts their score, they should promptly inform the instructor for correction.