Certainly! Below is a concise polynomial formula sheet summarizing key formulas, concepts, and properties related to polynomials.
Polynomial Formula Sheet
1. Definition of a Polynomial
A polynomial is an expression of the form: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where:
- \( n \) is a non-negative integer (degree of the polynomial),
- \( a_n, a_{n-1}, \ldots, a_0 \) are constants (coefficients),
- \( x \) is a variable.
2. Degree of a Polynomial
- The degree of a polynomial is the highest power of \( x \) with a non-zero coefficient.
- For example, \( P(x) = 4x^3 - 2x^2 + 7 \) has a degree of 3.
3. 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., \( P(x) = x^3 - x + 1 \)).
- Quartic Polynomial: Degree 4 (e.g., \( P(x) = x^4 + 3x^2 + 2 \)).
- Quintic Polynomial: Degree 5 (e.g., \( P(x) = x^5 - 5x^4 + 4 \)).
4. Polynomial Operations
- Addition: \[ (P(x) + Q(x))(a_n + b_n)x^n + (a_{n-1} + b_{n-1})x^{n-1} + \ldots \]
- Subtraction: \[ (P(x) - Q(x))(a_n - b_n)x^n + (a_{n-1} - b_{n-1})x^{n-1} + \ldots \]
- Multiplication:
- Distributive Law: \[ P(x) \cdot Q(x) = \sum a_i b_j x^{i+j} \]
- Division: Long division or synthetic division can be used.
5. Roots of Polynomials
- A polynomial \( P(x) \) has roots \( r_1, r_2, ..., r_n \) if: \[ P(r_i) = 0 \]
- Factor Theorem: If \( r \) is a root, then \( (x - r) \) is a factor of \( P(x) \).
- Rational Root Theorem: Possible rational roots are of the form \( \frac{p}{q} \), where \( p \) divides the constant term and \( q \) divides the leading coefficient.
6. Factoring Polynomials
- Common techniques:
- Factoring out the greatest common factor (GCF).
- Using special products:
- Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \)
- Perfect square trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \)
- Sum or difference of cubes:
- \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
- \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
7. Special Polynomial Identities
- Binomial Theorem: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k \]
- Newton's Sums: Relates power sums to the coefficients of the polynomial.
8. Descartes' Rule of Signs
- The number of positive real roots of a polynomial is equal to the number of sign changes in \( P(x) \) or less by an even integer.
- The number of negative real roots is equal to the number of sign changes in \( P(-x) \).
9. Polynomial Graph Behavior
- A polynomial of degree \( n \) can have at most \( n \) turning points.
- End behavior:
- If \( n \) is even and leading coefficient \( a_n > 0 \): rises left and right.
- If \( n \) is even and leading coefficient \( a_n < 0 \): falls left and right.
- If \( n \) is odd and leading coefficient \( a_n > 0 \): falls left, rises right.
- If \( n \) is odd and leading coefficient \( a_n < 0 \): rises left, falls right.
10. Synthetic Division
- Procedure for dividing a polynomial by \( x - c \) using coefficients:
- List coefficients.
- Bring down the leading coefficient.
- Multiply the root \( c \) by the value just written and add down the column.
Conclusion
This formula sheet provides an overview and quick reference to basic polynomial concepts, operations, and properties. For more advanced topics, further study in algebra or calculus may be necessary.
Feel free to reach out for any specific topic or additional details!
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