Lesson 3.4: Real Zeros of Polynomials

Rational Zeros of Polynomials

RationalZerosTheorem: If thepolynomial P(r) =anxn*an-$n-1 +.'. *a1x*ae hasinteger coefficients, then every rational zero of P is of the form:

rrhe corrthan* coeQQic\es\ oo\eadrnX coeQ$qrer*.

+ P ulhere P i1 a $ac\or o$- q ? ov it a Qac\ov . of rrhe

Example: P(r) = x3 '3x + 2Oo = L tac\ovt" \r1Oa=\ tac\ort \

potar\:\e xahqna\ ?.eYCIt' :\ )t'L

Example: P(r) = 2x3 + xz * lsx + 6ffq=b $achxtr'. \,trt,bfft = 'l_ tactoxt \,1pots*e\e ?eYs\'. t\.,\-?rk3rt h

HoW to Find the Rational Zeros of a Polynomial

1. List all possible teYos using tr," Sahona\ t.evot . theorem.

2. use S$;rtlndric drvr$\orn or lkre.Sencqt$0*ef \esve\c\to evaluite the polynomial at each of the candidates for the rational zeros that youfound in Step 1. When the remainder is 0, note the quotient you have obtained.

3. Repeat steps 1 and 2 until you reach a quotient that is a quadratic or factors easily.Use the quadratic formula or factor to find the remaining zeros.

?(1d = fr--1) ('l:x"+tx* %)

= (x-t) ("tx-t) (x+ t\

\er\ '

a1_

1\ -\e b

L E .T O

=) x- 1 \r a tac\or

?evot'. 'L ', t/r, -i

Entry #: l3-

q

Descartes' Rule of Signs

Let P be a polynomial with real coefficients.

1. The number of oot\h'of variations in sign in P(r) or

Lless than that by an even whole number.

zeros of f 1r) either is equal to the number

2. The number of neq3hrlQ \€3\ zeros af P(x) is either equal to thenumberof,,,i,@lessthanttriioyanevenwholenumber.

Example: Use Descartes' Rule to determine the possible number of positive and negativereal zeros of the polynomial P(r) = 3x6 * 4xi * 3r3 - r - 3.

Qot\\tve .

V(x\ = '3r(o +\xs + ixt -'A - i1, \, 3, -\ ) -3

\-1\ putr\.xe ?eru

-i\\ 0 -b L -i'i--i---**.U -t q -\t \%

%\ -u, b -\t ql

- t r5 \he \uu,')*rber\s\*,. &uv \\re?e,'{tt s\ t(x\

Neqa\rle -

pi*l= lxb-\xs -3xi +:x- 3j, -\, -3, \ '' -j\-r1 \-, \-,

i ov \ neqahue ?e.Y$%'

t\i 0 -t L -Eu1l- q 1- t\tr\a

L \t \he \l'VteYbound tur \\re?evuB nq qixl

t\

The Uooer and Lower Bounds Theorem

Let P be a polynomial with real coefficients.

1. lf we divide P(x) by * - b (with b > 0) using synthetic division and if the row thatcontains the quotient and remainder has no negative entry, then b is an upperbound for the real zeros of P.

2. lf we divide P(r) by * - o (with a < 0) using synthetic division and if the row thatcontains the quotient and the remainder has entries that are alternately nonpositiveand nonnegative, then a is a lower bound for the real zeros of P.

Example: Show that all the real zeros of the polynomial P(x) = x4 * 3xz + 2x - S liebetween -3 and 2.