Make a polynomial formula sheet

Answer 1

Here's a polynomial formula sheet covering key concepts, formulas, and properties related to polynomials:

Definitions

1. Polynomial: An expression of the form \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) where \( a_i \) are coefficients and \( n \) is a non-negative integer. The degree of the polynomial is \( n \).

2. Monomial: A polynomial with one term (e.g., \( 5x^3 \)).

3. Binomial: A polynomial with two terms (e.g., \( 3x^2 + 2x \)).

4. Trinomial: A polynomial with three terms (e.g., \( x^2 - 4x + 4 \)).

Basic Operations

1. Addition: \[ P(x) + Q(x) = (a_n + b_n)x^n + (a_{n-1} + b_{n-1})x^{n-1} + \ldots + (a_0 + b_0) \]

2. Subtraction: \[ P(x) - Q(x) = (a_n - b_n)x^n + (a_{n-1} - b_{n-1})x^{n-1} + \ldots + (a_0 - b_0) \]

3. Multiplication: \[ P(x) \times Q(x) = \sum_{k=0}^{m+n} \left( \sum_{i+j=k} a_i b_j \right) x^k \]

4. Division (using polynomial long division or synthetic division).

Special Formulas

1. Factoring:

   • Difference of Squares: \[ a^2 - b^2 = (a + b)(a - b) \]
   • Perfect Square Trinomial: \[ a^2 + 2ab + b^2 = (a + b)^2 \] \[ a^2 - 2ab + b^2 = (a - b)^2 \]
   • Sum/Difference of Cubes: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

2. Remainder Theorem: If a polynomial \( P(x) \) is divided by \( (x - a) \), the remainder is \( P(a) \).

3. Factor Theorem: \( (x - a) \) is a factor of \( P(x) \) if and only if \( P(a) = 0 \).

Properties

1. Degree of a Polynomial: The highest power of the variable \( x \).

2. Leading Coefficient: The coefficient of the term with the highest degree.

3. Zeroes of Polynomial: Solutions to the equation \( P(x) = 0 \).

Roots and Their Relations

1. If \( P(x) = a_n (x - r_1)(x - r_2)(x - r_3) \ldots (x - r_n) \):
   • Sum of Roots: \( r_1 + r_2 + ... + r_n = -\frac{a_{n-1}}{a_n} \)
   • Product of Roots (if \( n \) is even or odd):
     • If \( n \) is even: \( r_1 r_2 \ldots r_n = \left(-1\right)^n \frac{a_0}{a_n} \)
     • If \( n \) is odd: \( r_1 r_2 \ldots r_n = \left(-1\right)^{n+1} \frac{a_0}{a_n} \)

Polynomial Identities

1. Binomial Theorem: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

2. Newton's Binomial Expansion provided for expansion of \( (x + a)^n \).

Applications

• Polynomials are used in algebra, calculus (like Taylor and Maclaurin series), economics, engineering, and computer science (in algorithms).

Conclusion

This polynomial formula sheet provides a comprehensive overview of polynomials, their operations, properties, and applications. For more complex applications, one may explore numerical methods for roots, graphing techniques, and synthetic division in greater detail.