Example 6Draw 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. 245338dHmRxalmfCxlASameffxxlimPaka1mflx7awaaffectlocalbehaviornotend9gdkdifferentdifferentlimhakelimNakxiaxxlimhaka1amNltaxODDdegPolyofReiExtevenuponIfmustoccurinpairsmustbesmooth'scontinuousdegreeScannotdrawodddegevendegjµ2Textx.intcanhaveastndegpolymusthavenopolywitnox.intµcoppeBThffinat.tkastgfajgntee.ioisfkAtVEit4RelExtanevendegreenpolynomialorafunctioncanhavenn nbn2isoxinteIygWoAup9oddnffRelEAnodddegree npolynomial4rootsfunctioncanhave3RdExtn nbutIxintevennumberofRelextuptonl
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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 7Sketch a graph of the following function 232( )5 (3) (2) (5) (6)f xx xxxx±±²²±. State the leading term and the degree of the polynomial. A polynomial f(x) in factored form 23( )()() () ,f xA xaxbxc±±±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.) ()xa±is a factor of a polynomial function IFF x = a is a root of the same polynomial function. approachdozffetriplelikeyourootaregoingacrossmlm2m3crossbounceinfibouncecrossx ox3xzxsx6M1m2m3m2mldegree12 3249
Example 8Let 432( )23P xxxx±±². Find the zeros of P, then sketch the graph of P Example 9Let 32()248h mmmm±±². Find the roots of h using factor-by-grouping, then sketch the graph of h. Example 10Write 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) ³0,10´(ii) ³1, -4 ´HINT: It might help to sketch each first!
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