Question

Write a polynomial formula sheet like a 10th grader without any coding

Answer 1

### Polynomial Formula Sheet

**1. Definition of a Polynomial:**
A polynomial is an expression consisting of variables (often denoted as \(x\)) raised to non-negative integer powers, multiplied by coefficients, and added together. The general form of a polynomial in one variable \(x\) is:
\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0
\]
where:
- \(a_n, a_{n-1}, \ldots, a_0\) are coefficients (real numbers)
- \(n\) is a non-negative integer (the degree of the polynomial)

**2. Types of Polynomials:**
- **Constant Polynomial:** Degree 0 (e.g., \(P(x) = 5\))
- **Linear Polynomial:** Degree 1 (e.g., \(P(x) = 2x + 3\))
- **Quadratic Polynomial:** Degree 2 (e.g., \(P(x) = x^2 + 4x + 4\))
- **Cubic Polynomial:** Degree 3 (e.g., \(P(x) = x^3 - 2x^2 + x - 5\))
- **Quartic Polynomial:** Degree 4 (e.g., \(P(x) = x^4 + 3x^2 - 2\))
- **Quintic Polynomial:** Degree 5 (e.g., \(P(x) = x^5 + 0.5x - 1\))

**3. Polynomial Operations:**
- **Addition/Subtraction:**
Combine like terms.
\[
(a_n x^n + \ldots + a_0) + (b_n x^n + \ldots + b_0) = (a_n + b_n)x^n + \ldots + (a_0 + b_0)
\]

- **Multiplication:**
Use the distributive property (FOIL for binomials).
\[
(a + b)(c + d) = ac + ad + bc + bd
\]
For higher degrees, multiply each term in one polynomial by each term in the other.

- **Division (Polynomials):**
Use long division or synthetic division.

**4. Factoring Polynomials:**
Common factoring techniques include:
- **Factoring out the Greatest Common Factor (GCF):**
\[
ax^n + bx^n = (a + b)x^n
\]

- **Difference of Squares:**
\[
a^2 - b^2 = (a - b)(a + b)
\]

- **Perfect Square Trinomials:**
\[
a^2 + 2ab + b^2 = (a + b)^2
\]
\[
a^2 - 2ab + b^2 = (a - b)^2
\]

- **Quadratic Trinomials:**
\(ax^2 + bx + c\) can often be factored into \((px + q)(rx + s)\).

**5. Roots and the Factor Theorem:**
- A polynomial \(P(x)\) has a root at \(x = r\) if \(P(r) = 0\).
- If \(x - r\) is a factor of \(P(x)\), then \(P(r) = 0\).

**6. Remainder Theorem:**
- When a polynomial \(P(x)\) is divided by \(x - k\), the remainder is \(P(k)\).

**7. Graphing Polynomials:**
- The degree of the polynomial determines the number of turns in its graph:
- An even degree has endpoints that go in the same direction.
- An odd degree has endpoints that go in opposite directions.

**8. Key Values:**
- **Leading Coefficient Test:**
- If the leading coefficient is positive and the degree is even, the ends of the graph go up.
- If the leading coefficient is negative and the degree is even, the ends of the graph go down.
- If the leading coefficient is positive and the degree is odd, the left end goes down and the right end goes up.
- If the leading coefficient is negative and the degree is odd, the left end goes up and the right end goes down.

### Summary
Polynomials are fundamental components of algebra. Understanding their structure, operations, and characteristics is key for solving problems and performing advanced mathematical tasks.