Notes on Polynomial Long Division and Polynomial Functions

A polynomial function is a function that can be written in the form
P(x) = an x^{n} + a{n-1} x^{n-1} + \dots + a1 x + a0
The numbers a0, a1, \dots, a_n are called coefficients.
The degree of the polynomial is n (assuming a_n \neq 0).
The leading coefficient is an and the constant term is a0.
Standard form orders terms from highest power to lowest power.
Examples from the transcript (decoded):
- P(x) = 2x^3 + 5x^2 + 6x + 7
- f(x) = 7x + 4
- F(x) = 9x - 2
A quadratic is a polynomial of degree 2 with general form
f(x) = a x^{2} + b x + c, \quad a \neq 0
The transcript mentions many polynomials and coefficients in a garbled way; the following standard forms are used for practice and understanding:
- General cubic: P(x) = a3 x^{3} + a2 x^{2} + a1 x + a0
- General quadratic: f(x) = a x^{2} + b x + c

A polynomial function is defined by its polynomial expression; evaluating at a value gives the function's output.
Notation recap:
- Polynomial in x of degree n: an x^n + a{n-1} x^{n-1} + \dots + a1 x + a0
- Coefficients: a_i for i = 0,1,…,n.
- Leading coefficient: a_n
- Constant term: a_0
Examples extracted from the transcript (as readable forms):
- P(x) = 2x^3 + 5x^2 + 6x + 7
- p(x) = 6x + 3 (labeled as Pa in the garbled notes; clarified as a linear/axial example)
- P_r = 8x^2 + 12x + 4 (interpreted from garbled text as a quadratic example)
Conceptual notes:
- A polynomial’s shape and behavior are determined by degree and coefficients.
- Coefficients scale the contribution of each power of x.

The transcript includes several garbled lines. Some readable items and their interpretations:
- "Polynomial long division" topic is present.
- Examples that are clear:
- f(x) = 7x + 4
- F(x) = 9x - 2
- P(x) = 2x^3 + 5x^2 + 6x + 7
- The line "The numbers 90, 91, 9.x are Called Coefficients" appears to intend that coefficients are the numbers preceding each power of x, i.e., a0, a1, \dots, a_n.
- The garbled lines labeled as ex(amples) suggest there were practice polynomials to divide or factor, but specific, reliable forms are not preserved in the transcript.
Takeaway: rely on standard forms and the readable examples above when studying; note the ambiguities in the source transcription for later clarification.

Goal: divide a polynomial dividend by a polynomial divisor to obtain a quotient and possibly a remainder.
Steps (procedural):
- 1) Write dividend and divisor in standard form with descending powers of x.
- 2) Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient.
- 3) Multiply the divisor by that quotient term and subtract the result from the dividend.
- 4) Repeat the process with the new dividend (the remainder from the previous step) until the degree of the remainder is less than the degree of the divisor, or until no terms remain.
Final result can be written as:
\frac{\text{dividend}}{\text{divisor}} = Q(x) + \frac{R(x)}{\text{divisor}}
Key condition: if the remainder is zero, the divisor evenly divides the dividend; otherwise, the remainder is a nonzero polynomial with degree smaller than the divisor.

Divide the dividend D(x) = 2x^3 + 5x^2 + 6x + 7 by the divisor d(x) = x + 3.
Step 1: Leading term division
- Leading term of D: 2x^3; leading term of d: x
- Quotient term: 2x^2
Step 2: Multiply and subtract
- Multiply divisor by 2x^2: (x+3)\cdot 2x^2 = 2x^3 + 6x^2
- Subtract from D: (2x^3 + 5x^2 + 6x + 7) - (2x^3 + 6x^2) = -x^2 + 6x + 7
Step 3: Next quotient term
- Leading term of new dividend: -x^2; divide by leading term of divisor x to get -x
- Multiply divisor by -x: (x+3)(-x) = -x^2 - 3x
- Subtract: (-x^2 + 6x) - (-x^2 - 3x) = 9x; bring down constant term: 9x + 7
Step 4: Final quotient term
- Leading term: 9x; divide by x to get 9
- Multiply divisor by 9: (x+3)\cdot 9 = 9x + 27
- Subtract: (9x + 7) - (9x + 27) = -20
Result
- Quotient: Q(x) = 2x^2 - x + 9
- Remainder: R(x) = -20
Therefore:
\frac{2x^3 + 5x^2 + 6x + 7}{x + 3} = 2x^2 - x + 9 + \frac{-20}{x + 3}
Quick check: Multiply the quotient by the divisor and add the remainder to see the original dividend is recovered.
Practical note: If you start with a different arrangement of terms, you must maintain the same leading-term logic to avoid arithmetic mistakes.

Polynomials are sums of terms with nonnegative integer powers of x and coefficients.
The standard form orders terms by decreasing powers of x.
Coefficients determine each term’s contribution; the leading coefficient is the coefficient of the highest-degree term.
Polynomial long division mimics numerical long division: determine quotient terms one by one and subtract the multiplied divisor until a remainder with smaller degree than the divisor remains.
A remainder of zero indicates exact division; otherwise, the remainder can be written over the divisor in the final expression.

Several lines in the provided transcript are garbled and do not clearly translate into a specific mathematical statement (e.g., "Tineor FC)axtb", "Fax = x² = 6.15 fur - 13.116", and other ex-like fragments).
The readable, reliable parts include the general polynomial form, named coefficients, a specific polynomial example, a linear polynomial example, and a cubic example. These should be the focus when studying from this material.
If you have access to the original source (video/slides/pdf), consider re-checking the garbled sections for exact examples (especially any listed long-division or factoring problems) to ensure full coverage of those problems in your notes.

Write the standard form of a polynomial with degree 4 and coefficients a4, a3, a2, a1, a0.
- Express as: P(x) = a4 x^4 + a3 x^3 + a2 x^2 + a1 x + a_0
Given f(x) = 7x + 4 and F(x) = 9x - 2, identify the degree, leading coefficient, and constant term for each.
Perform polynomial long division of D(x) = 2x^3 + 5x^2 + 6x + 7 by d(x) = x + 3 and state the quotient and remainder.
If you divide a polynomial and obtain a remainder of zero, what does that signify about the divisor and dividend?