AP Precalculus: Polynomial Expressions & Equations Formulas
1. Polynomial Vocabulary
- Polynomial: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0 \)
- Degree: Highest power of \(x\)
- Leading Coefficient: \(a_n\)
- Constant Term: \(a_0\)
- Monomial, binomial, trinomial: 1, 2, or 3 terms
2. Polynomial Long Division
\[ \text{If } f(x) \div d(x) = q(x) \text{ with remainder } r(x), \text{ then } f(x) = d(x)q(x) + r(x) \]Arrange by descending degree, divide leading terms, repeat for result.
3. Synthetic Division
Use to divide by \(x - c\): write \(c\), bring down coefficients, multiply–add through.Shortcut for evaluating \( p(c) \) (Remainder Theorem).
4. Evaluate Polynomials (Remainder Theorem)
\[ p(c) = \text{remainder when dividing } p(x) \text{ by } x - c \]- Do synthetic division with \(c\). Final value = \(p(c)\).
5. Sum & Difference of Cubes
- \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
- \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)
6. Quadratic Pattern Factoring
For \( ax^{2n} + bx^n + c \):Let \( y = x^n \), factor as quadratic: \( a(y)^2 + by + c \)
7. Pascal's Triangle
Each row gives the coefficients for expanding \( (a + b)^n \) (nth row for power n).Row 0: 1 | Row 1: 1,1 | Row 2: 1,2,1 | Row 3: 1,3,3,1 | ...
8. Binomial Theorem
\[ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k \]- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) (binomial coefficient, from Pascal’s triangle)
- Each term: \( \binom{n}{k} a^{n-k} b^k \) for \( k = 0 \) to \( n \)