Question
Make a polynomial formula sheet
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GPT-4o mini
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.
### 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.