Monic Adjuncts of Quadratics (Developed, Composed & Typeset by: J B Barksdale Jr / )

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1 Monic Adjuncts of Quadratics (eveloped, Composed & Typeset by: J B Barksdale Jr / ) Article 00.: Introduction. Given a univariant, Quadratic polynomial with, say, real number coefficients, (0.1) QÐBÑ +B,B - ß there is an associated monic, quadratic polynomial, QÐBÑß the following, which is described by EFINITION: To each quadratic polynomial as in (0.1), there is an associated monic, quadratic polynomial ß QÐBÑß called the monic adjunct of UÐBÑß and which is defined by, (0.2) QÐBÑ B,B %+- monic adjunct of Q. As a consequence of some casual, algebraic, gymnastic doodling, several functional relationships (and connections) regarding the zeros of UÐBÑ and QÐBÑ did emerge. An account of such doodling--and the results there of--are below presented in the following sequel of Articles. Article 01.: Monic (Quadratic) Adjunct Polynomials. Consider a quadratic, UÐBÑß as in Item (0.1). Now, note that (1.1) %+ UÐBÑ %+ B %+,B %+- Ð+BÑ,Ð+BÑ %+-. So by declaring that +B; and, that QÐÑ be defined as (1.2) QÐÑ, %+-, the following relationships emerge. " (1.) (A): %+ UÐBÑ QÐ+BÑ and (B): UÐ Ñ QÐÑ Þ + %+ Appealing to (1.2), further exploration in such manner reveals that Ðfor Á!Ñ %+- "'+ -, %+- %+- %+- (1.4) QÐ Ñ QÐÑ. And then, from applications of Items (1.4) and (1.A), we have that (1.5) %+- % QÐÑQÐ Ñ QÐ+ Ñ%+UÐ Ñß - 1 -

2 from which we conclude that Ðfor Á!Ñ - - (1.6) UÐ Ñ QÐÑ Þ Items (1.B), (1.4) and (1.6) are now, hereby, organized together and conveniently displayed as a below boxed bundle of relationships in the spirit of convenience and ease of application. " %+- % (1.7) A: UÐ Ñ QÐÑ B: QÐ Ñ QÐÑ C: UÐ Ñ QÐÑ. + %+ Appealing to the relationships of (1.7), another interesting functional relationship presents itself. Hence, consider the following Theorem 1. Given a quadratic Uand its monic adjunct Qß suppose that that for ß A : A %+- Á!Þ Then: QÐAÑ AQÐÑ Þ Proof: Referencing Item (1.7B), note thatß A %+- %+- A A QÐAÑQÐ Ñ QÐ Ñ QÐÑ QÐÑ QÐÑÞ Therefore, it now follows that, (1.8) QÐAÑ AQÐÑ Þ Article 02.: Novel Curiosities of Q and M Pairs. By implementing the functional relationships between a quadratic U and its monic adjunct Qß novel relationships involving the zeros of such Uand Q polynomial pairs can be developed. We begin with the following theorem demonstrations. Theorem 2. Given a quadratic Uand its monic adjunct Qß suppose that for ßA : A%+-Á!Þ Then: QÐÑ! Í QÐAÑ!Þ Proof: Given the hypothesis, (1.8) is applicable. Also, Á!ÁAÞHence, by (1.8) À AQÐÑ! Í QÐAÑ! Þ Thus, we have that, QÐÑ! Í AQÐÑ! Í QÐAÑ! Í QÐAÑ!Þ Therefore: QÐÑ! Í QÐAÑ!Þ COROLLARY 2.1 If A %+-! for two non-zero values, z,w, then: Either both are zeros of Qß or else, Neither is a zero of M. Zeros of a quadratic polynomial ßUßand those of its monic adjunct, Qßcan now be related to each other by applying--in concert with each other--the results of Item (1.7A) together with Theorem

3 Theorem. Given a quadratic Uand its monic adjunct Qß suppose that - QÐÑ! for where Á!. Then: UÐ Ñ! UÐ Ñ Þ + Proof: Given that QÐÑ! and Á!, and then appealing to Items (1.7A) " - - and (1.7C), it follows that: UÐ Ñ QÐÑ! QÐÑUÐ Ñ. + %+ COROLLARY. Given a quadratic (polynomial) UÐBÑ, suppose that Á!ßand that is a fixed zero of the monic adjunct QÐBÑÞ Then, zeros of UÐBÑ are given by the formulas (2.1) - B" and B + ß Ð Á!Ñ. Theorem 4. Given a quadratic Uand its monic adjunct Qß suppose that QÐÑ! for where Á!. Then, the zeros of UÐBÑ given by the formulas in Item (2.1) are distinct zeros iff Á,Þ Proof: We proceed to demonstrate that: B" B Í,Þ Hence, for Á! and QÐÑ!ßsuppose that B" B ÞThen, - B" B Ê + Ê %+-! Þ Also, since QÐÑ!ß, %+-!Þ Adding equations in, we arrive at!,ð,ñþgiven Á!ßit we conclude,þ Consequently, the implication: B" B Ê, follows. Conversely, now suppose that,þthen, we see that!qðñqð,ñð,ñ,ð,ñ %+-, %+-Þ Thus,, Ê, %+-! Ê, - +, Ê B" BÞ Therefore, we have established: B" B Í,. Hence, the inverse biconditional: B Á B Í Á, can be concluded. " Theorem 5. Given a quadratic (polynomial) UÐBÑ, suppose that Á!ßand that is a fixed zero of the monic adjunct QÐBÑÞ Then, zeros of UÐBÑ, B" and B ß given by the formulas of Item (2.1) satisfy the following equalities: (2.2), - (i) B" B and (ii) B " B + + Þ Proof: Given that QÐÑ! and Á!, and observing that!qðñ, %+- Ê %+-,Þ - %+-,, Hence, B" B Þ - - And, also, B B Þ "

4 In order to present the developments (and observations) of the preceding Articles as a conveniently organized list of itemized results, a summary of such results is now displayed below. Summary of Established Results and Observations: (2.) For and Á! : is a zero of Q iff is a zero of + %+- (2.4) For and Á! : is a zero of Q iff is a zero of UÞ Q Þ (2.5) For and Á!ß suppose that is a zero of Q, then: ÐA: Ñ and - are zeros of U (B): istinct zeros iff Á, Þ + (2.6) For and Á!ß suppose that is a zero of Q; then, the zeros of Uß B" and B, displayed in Item (2.1) satisfy the equalities (i) B B, and (ii) B B - " " Þ + + (2.7) Note that the condition, %+- Á!, restricts U from having zero values for either the leading coefficient, a, or the constant term, c. Of course, the condition that + Á! is implicit; otherwise, Uwould not be a quadratic. Observe that -Á! excludes the instance of Uwith a zero which has a zero value. (2.8) The condition that, - Á! would restrict the admittance of the trivial Uß +B!Þ Hence, exclusion of such trivial U can be assured by considering only admissible quadratic (polynomials) U as defined by the following (2.9) EFINITION: egenerate and trivial quadraticsßußcan be excluded by considering only admissible quadratics, Uß which satisfy the condition that: + Ð, - ÑÁ!Þ Article 0.: Alternative Renditions to the Quadratic Formula. By appealing to the results of the preceding Articles, novel and alternative renditions to the standard quadratic formula can be developed and formulated. In order to accomplish this objective, we introduce the following expression references. EFINITION: Given a general quadratic equation, UÐBÑ! ß the expressions created by the plus and minus sign options in the formulaß (.1) I,, %+- ß (root effectors formula) are hereby defined as quadratic root Effectors

5 (.2) We now note the following observations regarding quadratic root effectors: (i) Hegenerate and trivial quadratics are avoided by considering only admissible forms which satisfy the condition: + Ð, - ÑÁ! (as detailed in Item (2.9) ). (ii) IÁ!, for at least one of the root effector options (for admissible forms ). Theorem 6. The roots of an admissible quadratic equation, UÐBÑ!ß can be determined by using the following quadratic root effector formulas ÐUVIJ Ñ with a non-zero root effector. I - (.) B" and B Þ ÐUVIJ Ñ + I Proof : From (.2) we see that there does exist at least one non-zero root effector option of that admissible Q. Further, we note that for the monic adjunct, Qß we have that QÐBÑB,B, Ð, %+-Ñ ÐB,Ñ Ð, %+-Ñßandß consequently, QÐ IÑÒÐ,, %+- Ñ, Ó Ð, %+-Ñ!ß for either option of the root effectors. Now, by applying an I Á! option, Ð IÑ Á! and QÐ IÑ!, together with (2.1) of Theorem, establishes Item (.), and so, completes this proof. Theorem 7. The roots of an admissible quadratic equation, UÐBÑ!ß which are formulated by the root effector formulas of Item (.) satisfy the equalities, - (.4) (i) B" B and (ii) B " B Þ + + Proof: Here, we simply apply Theorem 5 where: Ð IÑÁ! that QÐ IÑ! as developed in the proof of Theorem 6. and observing Having observed the fundamental relationships and connective interplay between a quadratic U and its monic adjunct M as presented in the preceding Articles, this next result should actually come a as no surprise. Theorem 8. The roots of an admissible quadratic equation, UÐBÑ!ß as determined by quadratic root effector formulas of Item (.), can be classified by the monic adjunct discriminant M Ð,Ñ as follows: (.5) ÐÑ QÐ,Ñ! Ê ÐÑ QÐ,Ñ! Ê ÐÑ QÐ,Ñ! Ê distinct, real roots one, real root complex conjugate roots Proof: Note that QÐ,Ñ Ð,Ñ,Ð,Ñ %+-, %+- Þ Now, by referencing the root effectors formula in Item (.1) and observing that since QÐ,Ñ is, in fact, the radicand thereof, then the parity and value of QÐ,Ñ clearly classify the roots of UÐBÑ! as described in Item (.5) Þ - 5 -

6 QM_Ex01_02.nb 1 EXAMPLE 01 REGARING QUARATIC AJUNCTS H Now consider the Given Quadratic: Q HxL = x 2 5 x+2. Then, M HxL = x 2 10 x+24. Now proceed with the solution process. The symbol "E " is Protected. Hence, we use "R" to designate E L Clear@a, b, c, R, x1, x2 Q@x_ := ax 2 + bx + c M@x_ := x 2 10 x + 24 a = ; b= 5; c = 2; SequenceForm@ "MH bl = ", M@ b, " < 0 " SequenceFormA "R ε ", 9 b + è b 2 4 ac, b è b 2 4 ac =E MH bl = 1 < 0 R ε 8 4, 6< MH bl = 1 < 0 H Monic iscriminant < 0 ==> Two Real Zeros L H Select R = 4, and then calculate x1 and x2 R = 4; SequenceFormA "x1 = ", H RL 2 a, " and ", "x2 = ", 2 c H RL E L x1 = 2 and x2 = 1 H Checking these x values,... we find that L SequenceFormA "QHx1L = ", QA 2 E, " and ", "QHx2L = ", Q@1E QHx1L = 0 and QHx2L = 0 QHx1L = 0 and QHx2L = 0 H Illustration_ 01 is COMPLETE L H ================================================================= L EXAMPLE 02 ILLUSTRATIVE ETAILS PRESENTE ON NEXT PAGE - 6 -

7 QM_Ex01_02.nb 2 EXAMPLE 02 REGARING QUARATIC AJUNCTS H Now suppose that:q HxL = 9 x 2 12 x+5. Then, M HxL = x 2 24 x+180. Proceeding with symbol "R" to designate E, we have that... L Clear@a, b, c, R, x1, x2 Q@x_ := ax 2 + bx + c M@x_ := x bx+ 4 ac a = 9; b= 12 ; c = 5; SequenceForm@ "MH bl = ", M@ b, " > 0 " SequenceFormA "R ε ", 9 b + è b 2 4 ac, b è b 2 4 ac =E MH bl = 6 > 0 MH bl = 6 > 0 H Monic iscriminant > 0 ==> Complex Zeros L R ε I, 12 6I< H Select R = 12+6 I, and then calculate x1 and x2 R = I; L SequenceFormA "x1 = ", H RL 2 a, " and ", "x2 = ", 2 c H RL E x1 = 2 I and x2 = 2 + I H Checking these x values,... we find that L SequenceFormA "QHx1L = ", QA 2 I 2 E, " and ", "QHx2L = ", QA + I EE QHx1L = 0 and QHx2L = 0 QHx1L = 0 and QHx2L = 0 H Illustration_ 01 is COMPLETE L - 7 -

8 QM_Ex0.nb 1 EXAMPLE 0 REGARING QUARATIC AJUNCTS H Imagine a Quadratic Polynomial Q HxL over the Finite Field of integers Modulo H1L, GF H1L, and denoted by the symbol Z 1. This illustration computes the ZEROS of the Given Quadratic, Q HxL = 12 x 2 +5 x+9, over Z 1, by implementing the methods of this composition. ===================================================================== In the spirit of convenience,... both: HSqrListL the Squares of Z 1 and, also, HInvListL the Multiplicative Inverses of Z 1, are hereby listed in order of: SqrList = 81 2 through to 1 2 <; AN InvList = 81 1 through to 1 1 <, respectively. Now,... consider the following results from Mathematica. Note: The Symbol "E" is PROTECTE in Mathematica; here, the Symbol "R " will INSTEA be used AS A SUBSTITUTE for the SYMBOL "E " in what follows here. L SqrList = PowerMod@81, 2,, 4, 5, 6, 7, 8, 9, 10, 11, 12<, 2, 1; InvList = PowerMod@81, 2,, 4, 5, 6, 7, 8, 9, 10, 11, 12<, 1, 1; SequenceForm@"SqrList = ", SqrList SequenceForm@ "InvList = ", InvList H And, so... L SqrList = 81, 4, 9,, 12, 10, 10, 12,, 9, 4, 1< InvList = 81, 7, 9, 10, 8, 11, 2, 5,, 4, 6, 12< ETAILS & EVELOPMENT CONTINUE ON NEXT PAGE - 8 -

9 QM_Ex0.nb 2 H Applying the Results of this Composition to Q HxL, below, note that L Q@x_ := 12 x x + 9 Clear@a, b, c, R, x1, x2 a = 12; b = 5; c = 9; SequenceFormA "R ε ",9 b + è Mod@ b 2 4 ac, 1, b è Mod@ b 2 4 ac, 1=E H Hence, we observe that L R ε 88, 2< H Therefore, applying the Root Effector Formulas by selecting R = 8, L R = 8; x1 = ModA H RL 2 c, 1E; x2 = ModA 2 a H RL, 1E ; SequenceForm@ "x1 = ", x1, " and ", "x2 = ", x2 H And so, we have that... L x1 = 8 and x2 = 4 4 x1 = 8 and x2 = 4 4 H So that Mod H1L, x1 = 8 1 and x2 = L Clear@x1, x2 x1 = Mod@8 9, 1; x2 = Mod@4 10, 1 ; H Appealing the "InvList " above L SequenceForm@ "x1 = ", x1, " and ", "x2 = ", x2 H Consequently it follows that... L x1 = 4 and x2 = 1 SequenceForm@ "QH4L = ", Mod@Q@4,1, " and ", "QH1L = ", Mod@Q@1, 1 H These x values ARE ZEROS via the above Mathematica Code... L QH4L = 0 and QH1L = 0 H Hence, THIS EXAMPLE 0 IS COMPLETE... L - 9-