5. Characteristics of Polynomials Review 2024

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$2x^3 + 3x^2 - 5x + 7$$ is a polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$4x^5 + 3x^2 - 2$$ , the degree is 5.

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. They can be found using factoring, the quadratic formula, or numerical methods.

A quadratic polynomial is a polynomial of degree 2, typically in the form $$ax^2 + bx + c$$ , where a, b, and c are constants and @@a eq 0@@.

Even degree polynomials have the highest power as an even number and their graphs are symmetric about the y-axis. Odd degree polynomials have the highest power as an odd number and their graphs are symmetric about the origin.

The graph of a polynomial function is continuous and smooth, with no breaks or sharp corners. The number of turning points is at most one less than the degree of the polynomial.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in $$3x^4 + 2x^3 - x + 5$$ , the leading coefficient is 3.

The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root. A polynomial of degree n has exactly n roots, counting multiplicities.

The number of roots of a polynomial can be determined by its degree. A polynomial of degree n can have up to n real roots.

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form $$(x - c)$$ . It is faster than long division and is used to find roots.