1. POLYNOMIALS A polynomial is a sum or difference of monomials (terms). Two terms = binomial Three terms = trinomial E VALUATING P OLYNOMIAL F UNCTIONS

2. A polynomial function is a function of the form f (x) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n  0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n  0 anan anan leading coefficient a 0a 0 a0a0 constant term n n degree descending order of exponents from left to right. n n – 1

3. DegreeTypeStandard Form E VALUATING P OLYNOMIAL F UNCTIONS You are already familiar with some types of polynomial functions. Here is a summary of common types of polynomial functions. 4Quartic f (x) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 0Constantf (x) = a 0 3Cubic f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 2Quadratic f (x) = a 2 x 2 + a 1 x + a 0 1Linearf (x) = a 1 x + a 0

4. 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 – 3x 4 – 7 1 2 S OLUTION The function is a polynomial function. It has degree 4, so it is a quartic function. The leading coefficient is – 3. Its standard form is f (x) = – 3x 4 + x 2 – 7. 1 2

5. 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. Identifying Polynomial Functions 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. S OLUTION f (x) = x 3 + 3 x

6. 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. S OLUTION f (x) = 6x 2 + 2 x – 1 + x The function is not a polynomial function because the term 2x – 1 has an exponent that is not a whole number.

7. 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. S OLUTION The function is a polynomial function. It has degree 2, so it is a quadratic function. The leading coefficient is . Its standard form is f (x) =  x 2 – 0.5x – 2. f (x) = – 0.5 x +  x 2 – 2

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

9. G RAPHING P OLYNOMIAL F UNCTIONS 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.”

10. G RAPHING P OLYNOMIAL F UNCTIONS END BEHAVIOR

11. G RAPHING P OLYNOMIAL F UNCTIONS END BEHAVIOR FOR POLYNOMIAL FUNCTIONS C ONCEPT S UMMARY > 0even f (x)+  f (x) +  > 0odd f (x)–  f (x) +  < 0even f (x)–  f (x) –  < 0odd f (x)+  f (x) –  a n n x –  x + 

12. x f (x) –3 –7 –2 3 –1 3 0 1 –3 2 3 3 23 Graphing Polynomial Functions Graph f (x) = x 3 + x 2 – 4 x – 1. S OLUTION 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) + as x +.

13. x f (x) –3 –21 –2 0 –1 0 0 1 3 2 –16 3 –105 The degree is even and the leading coefficient is negative, so f (x) – as x – and f (x) – as x +. Graphing Polynomial Functions Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. S OLUTION 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.

14. Graphing Polynomial Functions Even and Odd Symmetry An EVEN graph is symmetric about the y-axis. An ODD graph is symmetric about the origin.

15. 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.

16. Polynomial in standard form Using Synthetic Substitution 2 x 4 + 0 x 3 + (–8 x 2 ) + 5 x + (–7) 2 6 6 10 18 35 30105 98 The value of f (3) is the last number you write, In the bottom right-hand corner. The value of f (3) is the last number you write, In the bottom right-hand corner. 20–85 –720–85 –7 Coefficients 3 x -value 3 S OLUTION Polynomial in standard form