Ers betweenctangle that isnd proceed by

ptential

l"

igrrre 5= ,tix3 I 4rx2 andy : 169

as potential ItfI Exercises

i

,.- ? ,lhrs twoZooming in.

!-3 and 1.3. I

bemisphericalr". *'hat is then?

-*{ I List all possible rational zeros given by the Rationallx-:s Theorem (but don't check to see which actually are zeros).

*Fl-r) :x3-4x2+3\ok):xo-3x3'6x+8i. frr.r) : zxs + 3x3 + 4x2 - 8

+ -rr) :6xa-x2+2x+;12,l^7: f .r) : 4x" - 2x'- 7

r" L'-r) : lZxs * 6x3 - 2x - 8

i*lr$ r A polynomial function P and its graph are given.,m List all possible rational zeros of P given by the Rational

Zeros Theorem.lrr From the graph, determine which of the possible rational

l;ros actually tum out to be zeros.

-1 1-- ---..) :5x'-x--5x*l

er of this boot

ilt.f,r):3x)+4x2-x-2

SECTION 3.3 Real Zeros of Polynomials 279

to graph the function , : !rx3 * 4rrx2, as shown in Figure 5. Since we want thevalue of this function to be 100, we also graph the horizontal line y : 100 in thesame viewing rectangle. The correct radius will be the x-coordinate of the point ofintersection of the curve and the line. Using the cursor and zooming in, we see thatat the point of intersectiot x : 2.15, coffect to two decimal places. Thus, the tankhas a radius of about 2.15 ft. r

Note that we also could have solved the equation in Example 8 by first writing it as

\nr3+4rr2-1oo:oand then finding the r-intercept of the function , : !rx3 * 4nx2 - 100.

9. P(x) : 2xa - 9x3 + 9x2 * x - 3

: rolume is

I equation:

*ion. and b1,ft-'il volum-dius lies som-3lb1'-s0. l!01

f "liit, +f-.

'l

v

10. P(x) :4xa - xj - 4x + I

v

11-40 r Find all rational zeros of the polynomial.

ll. P(x) = x3 + 3x2 - 4

12. P(x): x3 - 7x2 + l4x - 8

13.P(x):x3-3x-2A.P(x):x3+4x2-3x-1815. P("r) : x3 - 6x2 + 12x - 8

16.P(x):x3-x2-8x+1217.P(x):x3-4x2+x+618. P(x) : x3 - 4x2 - 7x + l019. P(x): x3 + 3x2 + 6x + 4

l|'-rii

I ols eq1 JoDq3as ul?urp aulnlo^ 'ii

orlecllddv

0 Jeq1le eABq

o.(Pcuxe seq

-_r4rerD aogs z5

B segsllsss segsllssord o1 urerrg eq1 eso (')1tr3Joeql

S aql e^ord (q)

Jr3r+ n'oqs (e)

ffI'3,\rle8euuoue I rEql esoddn5

)dg aqr aso'0 < q

ipdr aq (x)7 te'r

i-'o-rf00'z+sf- ir00't t rf,O0'f

-16+ zx\-txzO=l-x-Yx

oill ol laeuoc 'uol

tsqdef B esrl r

f-r9+zT8+€r€*:rfl- exS.-#Z

r:0=?*zx|-Yx.f,1 +rV-zxL-txteos eq uec suollnlos

Rt, 3ururJelep ol

pq$era ueql Pue

t-puonn, elqrssod 1e

Eoqnlos per eqJ' I

- r - -r = (x)aL-* t

9 + x9€ - txLS + exZZ - $tI - 14 : (r)7 .94

tZ + x1g - zxtZ * rx6 + fL - Ex : (r)a .UxS - zx3 - rxL - oxg : (x)4 '97

g + z1IZ - +xt : (r)a 'StV + xOZ t ,xI€. * ErgI + ,x7 : (x)4 'tr1

Z + xa _ zxl - JE t rxz _ (*)a .gl'senbruqoel Suuolceg: reqlo

ro 'ElnurroJ crle;penb eql 'su8r5;o elng .seUucseq 'ueroeqlspunog re.^Ao'I puu.reddn eqt 'ureroeqJ solez Ieuoue5 eqlesn 'eleudordde re^eueq^['fue;r 'sorez Izuorlexr eql pug

ueql pue '1enuou,(1od eqt Jo sorez leuorler IIB puld . 9L-EL

l-rx-sx:(y)d'ZLZ + x6 - rx I exZ - rx : (r)a lt

ZI + xg - zxE - ,x7: (x)a .0L

I + ,xg - tx : (')a 'eS'1eurou,(1od eql Jo soJez IeeJ

:qt.roJ spunoq re,^Aol pue;eddn ere 1eq1 sre8elur p]oilg. . ZL-69

9: Q'0 --D :l + )% - zxlZ * JLI - org: (r)a .g9

Z: q't- : D :6 + x6t - zr0l + ,xg : (x)a 'L9

9 : q'E- = D :B + xZ t zx6 - JZ - f : (r)a 'SS

I : Q't- - D tZ - r + zxS * txZ : (r)a'SS'prruou,{1od eqi Jo sorez Ieer eql ro; spunoq reddn

ru? Je.lAol eru q pue t2 JoJ senlul ua,Lr8 eql leql ,roqs r g9-S9

I + x* r.x t tx - f + Ex - sx : (r)a .VS

x9]_zx-fi+rr:(r)a'S9Zl + x * rx + tx t rx : (*)a .ZS

I _ xg _ r.x _ oxg + exz : (r)a .fS

L * xv + zx _ JZ: (r)a .os

t-x-zx-tx:(")a'OS'sorez IeeJ Jo Jequnu p1o1 elqrssod eql eururrelep ueqJ 'e^erl

-r prurou,(1od eql sorez per elrlu8eu .(ueru .uoq pue elrlrsod,ruzur .uoq eurturelep ol su3r5;o elnu ,seu?rsac asn r ?g-69

€ + rII - zxtl + f9 - rx - sx : (")a 'SS

t + x8 + zx t J9 - ox - ,x: (x)a '$8 - 18 + zr0l + ,x- : (x)4 '99

B - xl t zx9 + J9 - vx : (")a'SS6 _ rIZ I zrLl + ,xg: (x)a .V9

t + xV t ,xL - ,x7: (x)a .€9

9 + x9 t ,xZ - ,x- : (x)4 .79

Zl + xv _ zxE _,r: (r)a .IS

'ggo qde;8 eqt qrle{S (q)'drJo sorez Iuer eql 1e purg (e)

'ue,r.r8 sr g lerurou,(1od V r SS-IS

6 + xO9 * zxt6 t J9 * rr8l - ,ry : (x)4 .gg

I - r€ + zxLI + ErgI + ox7: (x)4 .6p

Z * xB - zx9 - Jt: (")a .St

I + rr9 - fi: (r)a 'nI - xZ* zr0l + J - vxl - Ex : (*)a .SV

6 - xt - zxtr + JL * { : (*)a .St

Z + xt - zxZ - {Z + rx : (r)a .W

t+x;I*zxt+J9-$:@)a.*Zt + xZ t zx9 - ,x : (x)4 '7p

Z-xt+zxt*J:@)A.W'(e)g elduruxg ur se ',(ressaceu Jr elnruJoJ crle.rpenb

eqt esn 'purou,(1od eqt Jo sorez per eqt rc puld r oS-LZI - xzt t ,xLZ - J6Z t vxel - sxE - sxZ: (")a .Ot

OI + xtt t zxg; t JtI - $tt - ,rg : (r)g .69

VZ _ xt _ zxZZ * rrt _ fi _ sx : (")a .SC

9E + ,x1g * J6 - ,xE * Er : (x)d 'LE

Z + xE + zxZI - JL - .rr9: (")a .Se

V - rB + zxt t JL * {Z : (")a .Se

Z - xE - zxll + ,x9 : (r)4 .yg

gI _ xU _ zrg + ,xy : (x)a .Ee

t - x - zxrgl + ,xg: (x)a .Ze

t + xL _ ,xy : (x)a .ttt + xZ - zxf_ - ,x7: (x)4 'gg

I-x-zxttJt:(*)a'eZV - xV * zxL + ,x7: (x)4 'g7

gI + xZE * zrlz * sxg + ,x : (x)l .Lz

9 + r€ + zxg _ J _ vx : (r)a .SZ

gE + zxgz - fi: (r)a 'sz

06 + r€ - zxtz - tx - rx : (x)4'Vg - 19 - zrL + rx9 + rx : (r)a .gZ

t * xB + zxt - JZ - rx : (r)a'CZ

t + ,x9 - rx : (-t)a 'fZ

E - rZ - zxz - ,r : (.r);' 'gg

suollcunJ leuorleu pue ;erurouA;od € Ull_dvH3 OgZ

.,fiod eq1 1"q1 ^doqs r

:.r:l

,L-t2 r Show thatthe polynomial does not have any. :al zeros.

,1' r- 1): X- - l - /,

- a.):2xa_ x3+x+2' P.):3x3-x2-6x+12,-- P,r) : ir5o - 5x25 + x2 - |t;*i{l r The real solutions ofthe given equation are rational.

' .11 possible rational roots using the Rational Zeros.:rem, and then graph the polynomial in the given viewing

, :rsle to determine which values are actually solutions.:olutions can be seen in the given viewing rectangle.)

,t ,:r - 3x2 - 4x + 12: O; l-a,+]by [-i5, 15]

* ,'' - 5x2 + 4 -- O; l-a,a)by [*30,30]i Lrl - 5x3 - t4x2 * 5;r * 12 = 0; l-2, 5]by [-40,40]

n,. -i.rj * g-r2 + 5x + 2:0; [-3,3] by[-10, 10]

'--9{} r Use a graphing device to find all reai solutions ofthe- j:ion, correct to two decimal places.

,'* ,:t-x-4:0,ln l.rr-8x2+9x-9:0,,, :.00x4 * 4.00x3 - 1o.g6x2 - 5.ggx + g.0g : 0,u ,:j * 2.OOxa + 0.96x3 + 5.00-12 + 10.00x + 4.g0:0' Let P(.r) be a polynomial with real coefficients and let

r ) 0. Use the Division Algorithm to write

P(,*):(r-b).Q(x)+rSuppose that r = 0 and that all the coefficients in Q("r) are:.onnegative. Letz) b.

a) Show thatP(z) > 0.,b) Prove the first part of the Upper and Lower Bounds

Theorem.c) Use the first part of the Upper and Lower Bounds Theo-

rem to prove the second part. fHint: Show that if P(r)satisfies the second part of the theorem, then P(-x)satisfies the first parl.l

'l Show that the equation

x5 - xo - x3 - 5x2 - 72x - 6: o

ras exactly one rational root, and then prove that it must:ave either two or four irrational roots.

rcplicationsVolume of a Silo A grain silo consists of a cylindricalnain section and a hemispherical roof. If the total volume-rf the silo (including the part inside the roof section) is

SECTION 3.3 Real Zeros of Polynomials 281

15,000 ft3 and the cylindrical part is 30 ft tall, what is theradius of the silo. correct to the nearest tenth of a foot?

Dimensions of a Lot A rectangular parcel of land has anarea of 5000 ft2. A diagonal between opposite corners ismeasured to be 10 ft longer than one side of the parcel.What are the dimensions of the land. correct to the nearestfoot?

Depth of Snowfall Snow began falling at noon onSunday. The amount of snow on the ground at a certainlocation at time / was given by the function

h(t) -* 1t.6ot - 12.4tt2 + 6.20t3

1.5811 +O.2ots-O.O1t6where / is measured in days from the start of the snowfalland h(t) is the depth of snow in inches. Draw a graph ofthis function and use your graph to answer the followingquestions.(a) What happened shortly after noon on Tuesday?(b) Was there ever more than 5 in. of snow on the ground?

If so, on what day(s)?(c) On what day and at what time (to the nearest hour) did

the snow disappear completely?

Volume of a Box An open box with a volume of1500 cm3 is to be constructed by taking a piece of cardboard20 cmby 40 cm, cutting squares of side length x cm fromeach comer, and folding up the sides. Show that this can be

BJ rttL'-'

95.

t.il

l,J

Ilt'

I

l, I

. l,

[:i: ,.::

j,, rr- :r-:, --

ll

h

J

96.