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.