n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 EVALUATING POLYNOMIAL FUNCTIONS A polynomial function is a function of the form f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 constant term an  0 an leading coefficient descending order of exponents from left to right. n n – 1 n degree Where an  0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a 0 is the constant term, and n is the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

You are already familiar with some types of polynomial EVALUATING POLYNOMIAL FUNCTIONS You are already familiar with some types of polynomial functions. Here is a summary of common types of polynomial functions. Degree Type Standard Form Constant f (x) = a 0 1 Linear f (x) = a1x + a 0 2 Quadratic f (x) = a 2 x 2 + a 1 x + a 0 3 Cubic f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 4 Quartic f (x) = a4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 2 – 3x4 – 7 1 2 SOLUTION The function is a polynomial function. Its standard form is f (x) = – 3x 4 + x 2 – 7. 1 2 It has degree 4, so it is a quartic function. The leading coefficient is – 3.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = x 3 + 3 x SOLUTION The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = 6x 2 + 2 x –1 + x SOLUTION The function is not a polynomial function because the term 2x –1 has an exponent that is not a whole number.

Identifying Polynomial Functions Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. f (x) = – 0.5 x +  x 2 – 2 SOLUTION The function is a polynomial function. Its standard form is f (x) =  x2 – 0.5x – 2. It has degree 2, so it is a quadratic function. The leading coefficient is .

f (x) = x 2 – 3 x 4 – 7 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x Identifying Polynomial Functions Polynomial function? f (x) = x 2 – 3 x 4 – 7 1 2 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x f (x) = – 0.5x +  x2 – 2

Using Synthetic Substitution One way to evaluate polynomial functions is to use direct substitution. Another way to evaluate a polynomial is to use synthetic substitution. Use synthetic division to evaluate f (x) = 2 x 4 + -8 x 2 + 5 x - 7 when x = 3.

Using Synthetic Substitution SOLUTION 2 x 4 + 0 x 3 + (–8 x 2) + 5 x + (–7) Polynomial in standard form Polynomial in standard form 2 0 –8 5 –7 3 x-value 3 • Coefficients Coefficients 6 18 30 105 2 10 35 6 98 The value of f (3) is the last number you write, In the bottom right-hand corner.

x +  is read as “x approaches positive infinity.” GRAPHING POLYNOMIAL FUNCTIONS The end behavior of a polynomial function’s graph is the behavior of the graph as x approaches infinity (+ ) or negative infinity (– ). The expression x +  is read as “x approaches positive infinity.”

GRAPHING POLYNOMIAL FUNCTIONS END BEHAVIOR

> 0 even f (x) +  f (x) +  GRAPHING POLYNOMIAL FUNCTIONS CONCEPT SUMMARY END BEHAVIOR FOR POLYNOMIAL FUNCTIONS > 0 even f (x) +  f (x) +  > 0 odd f (x) –  f (x) +  < 0 even f (x) –  f (x) –  < 0 odd f (x) +  f (x) –  an n x –  x +

f(x) Graph f (x) = x 3 + x 2 – 4 x – 1. Graphing Polynomial Functions Graph f (x) = x 3 + x 2 – 4 x – 1. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. The degree is odd and the leading coefficient is positive, so f (x) – as x and f (x) + . x f(x) –3 –7 –2 3 –1 1 2 23

f (x) Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. Graphing Polynomial Functions Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. The degree is even and the leading coefficient is negative, so f (x) – as x and f (x) + . x f (x) –3 –21 –2 –1 1 3 2 –16 –105