A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. $$ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 $$ Here, $x$ is the variable, $a_i$ are the coefficients, and $n$ is a non-negative integer representing the degree of the polynomial. • Terms: Each part of the polynomial separated by addition or subtraction is a term. For example, in $3x^2 + 2x - 5$, the terms are $3x^2$, $2x$, and $-5$. • Degree: The highest exponent of the variable in a polynomial is its degree. For $3x^2 + 2x - 5$, the degree is 2. • Monomial: A polynomial with one term (e.g., $5x^3$). • Binomial: A polynomial with two terms (e.g., $x + 2$). • Trinomial: A polynomial with three terms (e.g., $x^2 - 3x + 1$). Operations with Polynomials: 1. Addition/Subtraction: Combine like terms (terms with the same variable and exponent). Example: $(2x^2 + 3x - 1) + (x^2 - x + 4) = (2x^2 + x^2) + (3x - x) + (-1 + 4) = 3x^2 + 2x + 3$. 2. Multiplication: Use the distributive property. Multiply each term in the first polynomial by each term in the second. Example: $(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$. 3. Division: Can be done using polynomial long division or synthetic division. Factoring Polynomials: This is the reverse of multiplication. It involves finding polynomials that multiply together to give the original polynomial. Common methods include: • Factoring out the greatest common factor (GCF). • Factoring quadratics (e.g., $x^2 + bx + c$). • Difference of squares ($a^2 - b^2 = (a-b)(a+b)$). • Sum/Difference of cubes ($a^3 \pm b^3$).