A polynomial is an algebraic expression made up of terms which consist of variables raised to whole number exponents and their coefficients. Example: 4x² + 3x - 5.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in 5x³ + 2x² - x + 1, the degree is 3.

A quadratic polynomial is of degree 2 and is expressed as ax² + bx + c, where a ≠ 0. Example: 2x² - 3x + 4.

A cubic polynomial is of degree 3 and is expressed in the form ax³ + bx² + cx + d, where a ≠ 0. Example: 3x³ - 2x² + x.

Expressions involving negative or fractional exponents, division by a variable, or roots of variables are not polynomials. Example: 1/x or 2 + 3√x.

A zero of a polynomial p(x) is a value of x, say k, for which p(k) = 0. Example: In p(x) = x² - 4, the zeros are x = 2 and x = -2.

To find the value of a polynomial p(x) at x = k, replace x with k in the expression. Example: For p(x) = x² + 1, p(2) = 2² + 1 = 5.

To factor ax² + bx + c, find two numbers that multiply to ac and add to b. Example: x² - 5x + 6 factors to (x - 2)(x - 3).

The standard form is given by ax² + bx + c where a ≠ 0. Example: 3x² - 4x + 1 is in standard form.

Polynomial long division is a method used to divide polynomials similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor.

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form x - k. It is simpler than long division.

The graph of a polynomial represents the relationship between the variable and its output, showing the zeros, turning points, and end behavior.

For a quadratic ax² + bx + c, if the roots are p and q, then p + q = -b/a and pq = c/a.

A common mistake is confusing the coefficients or lower powers as the degree. Always identify the highest power of the variable.

A polynomial is linear if it has only one variable raised to the first power and no variable is multiplied together. Example: 4x + 3.

Polynomials are used in calculations related to areas, volumes, and in real-world problems involving rates of change.

Roots can be found using factoring, completing the square, or the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).