Degree n polynomial

Definition

A degree n polynomial is a mathematical expression in the form of $$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where the highest exponent of the variable x is n, and the coefficients $$a_n, a_{n-1}, ..., a_1, a_0$$ are real or complex numbers with $$a_n \neq 0$$. This type of polynomial plays a crucial role in understanding the roots and behavior of polynomial equations as well as their implications in fields like calculus and algebra.

5 Must Know Facts For Your Next Test

1. Every degree n polynomial has exactly n roots in the complex number system, counted with multiplicity, according to the Fundamental Theorem of Algebra.
2. If all coefficients of a degree n polynomial are real, then any complex roots must occur in conjugate pairs.
3. The graph of a degree n polynomial function can have at most n turning points, which are points where the graph changes direction.
4. A degree 0 polynomial is a constant function, while degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, and so forth.
5. Degree n polynomials can be factored into linear factors over the complex numbers, allowing for the complete solution of polynomial equations.

Review Questions

• How does the Fundamental Theorem of Algebra relate to degree n polynomials?
  • The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system. This means that for any degree n polynomial, we can expect to find all roots (real or complex) that satisfy the equation by finding solutions that make the polynomial equal zero. This connection highlights the importance of understanding degree n polynomials when analyzing polynomial equations.
• What implications do complex roots have for degree n polynomials with real coefficients?
  • When a degree n polynomial has real coefficients, any complex roots must appear as conjugate pairs. This means if one root is $$a + bi$$ (where i is the imaginary unit), there will also be a corresponding root $$a - bi$$. This property ensures that when factoring such polynomials, the factors remain with real coefficients, which is essential for graphing and analyzing the behavior of these polynomials on the real number line.
• Evaluate how the behavior of degree n polynomials changes as their degree increases and its impact on their graphical representation.
  • As the degree of a polynomial increases, its behavior becomes more complex. Higher degree polynomials can exhibit more turning points and can oscillate more frequently within a given range. For instance, while a linear (degree 1) function produces a straight line and quadratic (degree 2) functions create parabolas, cubic (degree 3) functions can have an 'S' shaped curve. This complexity not only affects how we interpret their graphs but also influences how we approach solving them algebraically and understanding their limits.

Related terms

Root: A root of a polynomial is a value of x for which the polynomial evaluates to zero.

Coefficient: A coefficient is a numerical factor in front of a variable in a polynomial expression.

Degree: The degree of a polynomial is the highest power of the variable x present in the polynomial.