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# Intro to Polynomials

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### Intro to Polynomials

1. 1. Polynomial Functions and Models <br />Module 12<br />
2. 2. Polynomial Functions<br />A polynomial of degree n is a function of the form<br />P(x) = anxn + an-1xn-1 + ... + a1x + a0<br />Where an 0. The numbers a0, a1, a2, . . . , an are <br />called the coefficients of the polynomial. <br />The a0is the constant coefficientorconstant term. <br />The number an, the coefficient of the highest <br />power, is the leading coefficient, and the term anxn is <br />the leading term.<br />
3. 3. Graphs of Polynomial Functions and Nonpolynomial Functions<br />
4. 4. Graphs of Polynomials<br />Graphs smooth curve<br />Degree greater than 2<br /> ex. f(x) = x3<br />These graphs will not have the following:<br />Break or hole<br />Corner or cusp<br />Graphs are lines<br />Degree 0 or 1<br /> ex. f(x) = 3 or f(x) = x – 5<br />Graphs are parabolas<br />Degree 2<br /> ex. f(x) = x2 + 4x + 8<br />
5. 5. Even- and Odd-Degree Functions<br />
6. 6. The Leading-Term Test<br />
7. 7. Finding Zeros of a Polynomial<br />Zero- another way of saying solution<br />Zeros of Polynomials<br />Solutions<br />Place where graph crosses the x-axis <br /> (x-intercepts)<br />Zeros of the function <br />Place where f(x) = 0<br />
8. 8. Using the Graphing Calculator to Determine Zeros<br />Graph the following polynomial function and determine the zeros.<br />Before graphing, determine the end behavior and the number<br />of relative maxima/minima.<br />In factored form:<br />P(x) = (x + 2)(x – 1)(x – 3)²<br />
9. 9. MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and:<br />K is even<br />The graph is tangent to the x-axis at (c, 0)<br />K is odd<br />The graph crosses the x-axis at (c, 0)<br />
10. 10. Multiplicity<br />y = (x + 2)²(x − 1)³<br /> Answer.  <br /> −2 is a root of multiplicity 2, <br />and 1 is a root of multiplicity 3.  <br />These are the 5 roots:<br />−2,  −2,  1,  1,  1.<br />
11. 11. Multiplicity<br />y = x³(x + 2)4(x − 3)5<br />Answer.   <br />0 is a root of multiplicity 3,<br />-2 is a root of multiplicity 4, <br />and 3 is a root of multiplicity 5.  <br />
12. 12. To Graph a Polynomial<br />Use the leading term to determine the end behavior.<br />Find all its real zeros (x-intercepts). <br />Set y = 0.<br />Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.<br />Find the y-intercept. Set x = 0.<br />Use any additional information (i.e. turning points or multiplicity) to graph the function.<br />