+ +
+--------+----- MtAml 511
I t
+
ASSIGNMENT 1 SEMESTER 1
-~11.L auestH1Nrs FlllLtL\il AN~WER~o +
t t t
DUE: MlRctH 2D26
DISCLAIMER:
THI' DOCUMEN IS FDR REFERENCE AND GUIDf'N E PURP SES
ONLY. I DD NOT TAKE RESPDNSIBIUITY !FDR AN~ PLAG ARl~M. ~MISUSE. DR
ACADEMIC MIS DNDUCT RESULTING FROM THSIR USE. IT IS YDURi j ~
===
RESPONSIBILITY TO ENSU~E ORIGINALITY AN~ CDMPL!IAN E WITH
- RELEVANT GUlr LINES DR t TAND, RDl . r l r
1
, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT
Question 1
(a) Flnd all possi ble ratifonal zeros of P(z)
To find the posstble ,atiot1al :zeros, ,_.,e use the Rational Root Theorem, which states that any
ratjonal zero rnLJiSt be in the form ::Le,
tj
where p is a factor of the constant term and q is a factor
of the leading coefficient
• Factors of t he const ant term (p = 4): -1=1, ...1::2 ±4
• Factors of the leading ccefHcient {q = 2): ± l.~±2:
Possib11e ra.tiona i zeros ( l!. ):
(J
Dividing each p by each q. we get:
(b) Use Oesca1
rtes' Ru le ct Signs to describe au possibTliUes
This rule !helps us narrow down the nature of the roots by looking at sign changes in the
coefficients.
1. Positive Re aI Ze rns: Check sign changes in P (x) = +2'x 1 - 7x 2 + 4x + 4.
• Frorn + 2 to - 7: 1 ~hange
• From -7 to + 4: 1 change
• Frorn +4 to + 4~ No change
• Total changes: 2. So, there are either 2 OJ" 0 positive tea! zero5.
2. Negati'\l'e Real Zeros: Oheck sign changes in P(-x}.
P(-x) = 2(-x)3 - 7(-~f + 4(-x) + 4 = - 2xl - 7x2 -4x + 4
• From - 2to - 7: No change
• From - 7 to - 4: No change
• Frorn -4 to + 4.~1 change
• Total rc hanges: 1. So, there is exactly 1 riegative rea I :zero.
+--------+----- MtAml 511
I t
+
ASSIGNMENT 1 SEMESTER 1
-~11.L auestH1Nrs FlllLtL\il AN~WER~o +
t t t
DUE: MlRctH 2D26
DISCLAIMER:
THI' DOCUMEN IS FDR REFERENCE AND GUIDf'N E PURP SES
ONLY. I DD NOT TAKE RESPDNSIBIUITY !FDR AN~ PLAG ARl~M. ~MISUSE. DR
ACADEMIC MIS DNDUCT RESULTING FROM THSIR USE. IT IS YDURi j ~
===
RESPONSIBILITY TO ENSU~E ORIGINALITY AN~ CDMPL!IAN E WITH
- RELEVANT GUlr LINES DR t TAND, RDl . r l r
1
, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT
Question 1
(a) Flnd all possi ble ratifonal zeros of P(z)
To find the posstble ,atiot1al :zeros, ,_.,e use the Rational Root Theorem, which states that any
ratjonal zero rnLJiSt be in the form ::Le,
tj
where p is a factor of the constant term and q is a factor
of the leading coefficient
• Factors of t he const ant term (p = 4): -1=1, ...1::2 ±4
• Factors of the leading ccefHcient {q = 2): ± l.~±2:
Possib11e ra.tiona i zeros ( l!. ):
(J
Dividing each p by each q. we get:
(b) Use Oesca1
rtes' Ru le ct Signs to describe au possibTliUes
This rule !helps us narrow down the nature of the roots by looking at sign changes in the
coefficients.
1. Positive Re aI Ze rns: Check sign changes in P (x) = +2'x 1 - 7x 2 + 4x + 4.
• Frorn + 2 to - 7: 1 ~hange
• From -7 to + 4: 1 change
• Frorn +4 to + 4~ No change
• Total changes: 2. So, there are either 2 OJ" 0 positive tea! zero5.
2. Negati'\l'e Real Zeros: Oheck sign changes in P(-x}.
P(-x) = 2(-x)3 - 7(-~f + 4(-x) + 4 = - 2xl - 7x2 -4x + 4
• From - 2to - 7: No change
• From - 7 to - 4: No change
• Frorn -4 to + 4.~1 change
• Total rc hanges: 1. So, there is exactly 1 riegative rea I :zero.
