Example 6 Draw several possibilities for the shape of a polynomial of even

# Example 6 draw several possibilities for the shape of

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Example 6 Draw several possibilities for the shape of a polynomial of even degree. Use a degree 4 polynomial as one such function. Analyze the possibilities for zeros and relative extrema. A polynomial function of degree n has at most n zeros and at MOST n -1 relative extrema. 2 4 5 3 3 8 d Hm Rx a lmfCxl A Same f f xx lim Pak a 1 mflx7 a w a affect local behavior not end 9 g d k different different lim hake limNak xia xx lim hak a 1am Nlt a x ODD deg Poly of ReiExt even up on I f mustoccurinpairs must be smooth's continuous degree S cannot draw odd deg even deg 2 Text x.int can have astndeg poly must have no polywitnox.int µ coppeBT hffinat.tkastgfajgntee.io is fk A tV Eit4RelExtanevendegree n polynomial or a function can have n n n bn 2 is ox int eIyg WoAup9oddn ffRelE An odd degree n polynomial 4 roots function can have 3RdExt n n but I x int even number of Relext upto n l

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In all the examples above, do you notice the relationship between the degree, the number of relative extrema, and the number of wiggles (inflection points)? Before we summarize what we’ve learned, we need to talk about repeated roots of polynomial functions. When a zero repeats itself d times, we say that zero has a multiplicity of d, or (md). The sum of the multiplicities of all the roots of a polynomial will equal the degree of the polynomial! In this class, you are responsible for identifying graphically, and from factored forms of polynomials, single roots (m1), double roots (m2), and triple roots (m3). Multiplicities in graphical and factored form A direct consequence of this is a very, very, very important theorem called the Factor Theorem . Factor Theorem Example 7 Sketch a graph of the following function 2 3 2 ( ) 5 ( 3) ( 2) ( 5) ( 6) f x x x x x x ± ± ² ² ± . State the leading term and the degree of the polynomial. A polynomial f(x) in factored form 2 3 ( ) ( )( ) ( ) , f x A x a x b x c ± ± ± will have roots of x = a (m1), x = b (m2) and x = c (m3). The degree of this polynomial will be 1 + 2 + 3 = 6 . A possible graph would look like this (of course, here, A would be a negative number.) ( ) x a ± is a factor of a polynomial function IFF x = a is a root of the same polynomial function. approach dozffe triple like you root are going across ml m2 m3 cross bounce infi bounce cross x o x 3 x z x s x 6 M1 m2 m3 m2 ml degree 1 2 3 24 9
Example 8 Let 4 3 2 ( ) 2 3 P x x x x ± ± ² . Find the zeros of P, then sketch the graph of P Example 9 Let 3 2 ( ) 2 4 8 h m m m m ± ± ² . Find the roots of h using factor-by-grouping, then sketch the graph of h. Example 10 Write the (a) general equation of the polynomial function whose only roots are x 5(m1), x ± 1(m2), x 2 (m3). Write the particular equations of the polynomial function from part (a) that passes through each the following points: (i) ³ 0,10 ´ (ii) ³ 1, -4 ´ HINT: It might help to sketch each first!

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