Polynomial functions are algebraic expressions with variables raised to non-negative integer exponents. This overview covers evaluating polynomials using direct substitution or synthetic division, understanding their graphical representation, determining zeros and y-intercepts, analyzing end behavior, and graphing techniques. Mastery of these concepts is crucial for algebraic studies and provides a foundation for further mathematical exploration.
Show More
Polynomial functions consist of terms with variables and coefficients
Polynomial functions are typically written in the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0
The leading term of a polynomial is the term with the highest exponent, and its coefficient is the leading coefficient
Polynomial functions can be evaluated by replacing the variable with a specific value and performing arithmetic operations
Synthetic division is a shortcut method for evaluating polynomial functions, especially useful for higher degree polynomials
All exponents in a polynomial must be whole numbers for it to be considered a polynomial
The degree of a polynomial influences the shape of its graph, with higher degree polynomials exhibiting more complex shapes
Zeros, also known as roots or x-intercepts, can be found using various algebraic techniques, and the y-intercept is found by evaluating the polynomial at x = 0
The end behavior of a polynomial graph is determined by the leading term and can be used to predict the long-term trends of the graph
Polynomial graphs are created by plotting points obtained through direct substitution and connecting them with a smooth, continuous curve
The leading coefficient test is used to determine the end behavior of a polynomial graph, which must be consistent with the polynomial's degree and leading coefficient
By combining information about a polynomial's zeros, y-intercept, and end behavior, an accurate graph can be sketched that captures its essential characteristics
00
In the expression f(x) = 3x^2 + 2x + 5, the coefficient of the term with the highest power is referred to as the ______ coefficient.
leading
01
Direct substitution method for polynomials
Replace x with specific value, perform arithmetic to find result.
02
Evaluating f(x) = 3x^2 + 2x + 5 at x = 4
Direct substitution: f(4) = 3(4)^2 + 2(4) + 5 = 61.
