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E VALUATING P OLYNOMIAL F UNCTIONS

E VALUATING P OLYNOMIAL F UNCTIONS. a n. n. n. n – 1. a 0. a n  0. leading coefficient. a n. constant term. degree. a 0. n. descending order of exponents from left to right. A polynomial function is a function of the form.

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E VALUATING P OLYNOMIAL F UNCTIONS

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  1. EVALUATING POLYNOMIAL FUNCTIONS an n n n– 1 a0 an 0 leading coefficient an constant term degree a0 n descending order of exponents from left to right. A polynomial function is a function of the form f(x) = an xn+ an– 1xn– 1+· · ·+ a1x + a0 Where an 0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a0 is the constant term, and nis the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

  2. EVALUATING POLYNOMIAL FUNCTIONS Degree Type Standard Form You are already familiar with some types of polynomial functions. Here is a summary of common types ofpolynomial functions. 0 Constant f (x) = a0 1 Linear f (x) = a1x + a0 2 Quadratic f (x) = a2x2+a1x + a0 3 Cubic f (x) = a3x3+ a2x2+a1x + a0 4 Quartic f (x) = a4x4 + a3x3+ a2x2+a1x + a0

  3. Identifying Polynomial Functions 1 f(x) = x2– 3x4– 7 2 1 Its standard form is f(x) = –3x4+x2 – 7. 2 Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is a polynomial function. It has degree 4, so it is a quartic function. The leading coefficient is – 3.

  4. Identifying Polynomial Functions f(x) = x3+ 3x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term 3xdoes not have a variable base and an exponentthat is a whole number.

  5. Identifying Polynomial Functions f(x) = 6x2+ 2x–1+ x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term2x–1has an exponent that is not a whole number.

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

  7. Identifying Polynomial Functions 1 f(x) = x2– 3x4– 7 2 f(x) = –0.5x+ x2– 2 Polynomial function? f(x) = x3+ 3x f(x) = 6x2+ 2x–1+ x

  8. Using Synthetic Substitution Use synthetic division to evaluate f(x) = 2x4 + -8x2 + 5x- 7 when x = 3. One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.

  9. Using Synthetic Substitution Polynomial in standard form 3 Coefficients x-value SOLUTION 2x4 + 0x3 + (–8x2) + 5x + (–7) Polynomial in standard form 2 0 –8 5 –7 3• Coefficients 6 18 30 105 35 10 98 2 6 The value of f(3) is the last number you write, In the bottom right-hand corner.

  10. The end behavior of a polynomial function’s graphis 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

  11. END BEHAVIOR GRAPHING POLYNOMIAL FUNCTIONS

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

  13. Graphing Polynomial Functions –3 –2 –1 0 1 2 3 x The degree is odd and the leading coefficient is positive, so f(x) –7 3 3 –1 –3 3 23 f(x) . and f(x) – – + + as x as x Graphf(x) =x3 + x2 – 4x – 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.

  14. Graphing Polynomial Functions x The degree is even and the leading coefficient is negative, so f(x) f(x) . and f(x) – – – + as x as x –3 –2 –1 0 1 2 3 –21 0 –1 0 3 –16 –105 Graphf(x) = –x4 – 2x3 + 2x2 + 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.

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