Polynomial Expressions and End Behavior

A polynomial expression is a mathematical expression that consists of variables raised to whole number powers, coefficients, and constants, combined using addition, subtraction, and multiplication. Example: $$2x^3 + 3x^2 - 5x + 7$$ .

The degree of a polynomial is the highest power of the variable in the expression. For example, in $$4x^3 + 2x^2 - x + 1$$ , the degree is 3.

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

End behavior describes how the values of a polynomial function behave as the input (x) approaches positive or negative infinity.

To find the end behavior, look at the leading term of the polynomial. If the leading term is positive and has an even degree, both ends go up. If it's negative, both ends go down. If the degree is odd, one end goes up and the other goes down.

As @@x ightarrow ext{-}\infty $$,$$ f(x) ightarrow ext{+}\\infty $$and as$$ x ightarrow +\infty $$,$$ f(x) ightarrow +\infty@@.

Polynomial long division is a method used to divide polynomials similar to long division with numbers, while synthetic division is a shortcut method used specifically for dividing by linear factors.

The result is $$3x^2 - 3x + 8 - \frac{9}{x+1}$$ .

The result is $$9x^2 + 9x + 5$$ .

The leading term determines the end behavior of the polynomial function as it dictates the direction of the graph as x approaches infinity or negative infinity.