4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x...

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4.4 Rational Root 4.4 Rational Root Theorem Theorem

Transcript of 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x...

Page 1: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

4.4 Rational Root 4.4 Rational Root TheoremTheorem

Page 2: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

Rational Root Theorem give direction in Rational Root Theorem give direction in testing possible zeros.testing possible zeros.

Let aLet a00xxnn + a + a11xxn-1n-1 + …a + …an-1n-1x + ax + ann = 0 represent a = 0 represent a

polynomial equation of degree n with polynomial equation of degree n with integral coefficients. If a rational number integral coefficients. If a rational number p/q, where p and q have no common p/q, where p and q have no common factors, is a root of the equation, then p is factors, is a root of the equation, then p is a factor of aa factor of ann and q is a factor of a and q is a factor of a00..

Page 3: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

Ex 1Ex 1List all the possible rational roots List all the possible rational roots then determine the rational roots.then determine the rational roots.

3x3x33 – 13x – 13x22 + 2x + 8 = 0 + 2x + 8 = 0

Page 4: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

Ex 2Ex 2Find ALL of the roots.Find ALL of the roots.

xx33 + 6x + 6x22 – 13x – 6 = 0 – 13x – 6 = 0

Page 5: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

Descartes’ Rule of SignsDescartes’ Rule of Signs

Used to determine the possible number of Used to determine the possible number of positive real zeros a polynomial has. positive real zeros a polynomial has. P(x) is a polynomial in descending order. The # P(x) is a polynomial in descending order. The # of positive real zeros is the same as the number of positive real zeros is the same as the number of sign changes of the coefficients or is less than of sign changes of the coefficients or is less than this by an even number.this by an even number.The # of negative real zeros is the same as the The # of negative real zeros is the same as the number of sign changes of the coefficients of P(-number of sign changes of the coefficients of P(-x), or less than by an even number.x), or less than by an even number.(Ignore zero coefficients.)(Ignore zero coefficients.)

Page 6: 4.4 Rational Root Theorem. Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial.

Ex 3Ex 3find the number of possible positive find the number of possible positive

and negative real zeros for then and negative real zeros for then determine the rational zeros:determine the rational zeros:

P(x) = 2xP(x) = 2x44 – x – x33 – 2x – 2x22 + 5x + 1 + 5x + 1


Plowing Through Sec. 2.4b with Two New Topics: Synthetic Division Rational Zeros Theorem.

Plowing Through Sec. 2.4b with Two New Topics: Synthetic Division Rational Zeros Theorem.

math1005postexam2reviewsolnF13 (1) Page 1quick049/Postexam2ReviewKEY.pdf5. Use the rational zeros theorem to determine the potential rational zeros Of the polynomial function f(x)

math1005postexam2reviewsolnF13 (1) Page 1quick049/Postexam2ReviewKEY.pdf5. Use the rational zeros theorem to determine the potential rational zeros Of the polynomial function f(x)

Sec 3.4 – Polynomial Functions Name: Rational Root Theorem ...gwinnett.k12.ga.us/.../unit03/03-04-RationalRtRemainderThrm-Compl… · The Remainder Theorem suggests that if a polynomial

Sec 3.4 – Polynomial Functions Name: Rational Root Theorem ...gwinnett.k12.ga.us/.../unit03/03-04-RationalRtRemainderThrm-Compl… · The Remainder Theorem suggests that if a polynomial

Day 11 - Rational Zero Theorem Notes.notebookerbenson.weebly.com/uploads/8/5/6/3/85639982/day... · Day 11 Rational Zero Theorem Notes.notebook 2 October 23, 2017. GSE Algebra 2 Rational

Day 11 - Rational Zero Theorem Notes.notebookerbenson.weebly.com/uploads/8/5/6/3/85639982/day... · Day 11 Rational Zero Theorem Notes.notebook 2 October 23, 2017. GSE Algebra 2 Rational

An explicit construction of spherical designsziqing.org/data/20171125 - An explicit construction... · Step 2. Rational-weighted rational semidesign on H1 1 Theorem 5 Assume that

An explicit construction of spherical designsziqing.org/data/20171125 - An explicit construction... · Step 2. Rational-weighted rational semidesign on H1 1 Theorem 5 Assume that

Polynomial, Power, and Rational · PDF filePolynomial, Power, and Rational Functions ... 2.4 Real Zeros of Polynomial Functions 2.5 Complex Zeros and the Fundamental Theorem of Algebra

Polynomial, Power, and Rational · PDF filePolynomial, Power, and Rational Functions ... 2.4 Real Zeros of Polynomial Functions 2.5 Complex Zeros and the Fundamental Theorem of Algebra

Look at Algebra 2 - Hanlon Mathy = –3 y = 2 now substitute and find x Higher Degree Equations - Rational Root Theorem 3 x+4x2+8x+5=0 by Rational Root Theorem ±1 ±1, ±5 ; ∴the

Look at Algebra 2 - Hanlon Mathy = –3 y = 2 now substitute and find x Higher Degree Equations - Rational Root Theorem 3 x+4x2+8x+5=0 by Rational Root Theorem ±1 ±1, ±5 ; ∴the

Find Rational Zeros, I Objectives: 1.To find the zeros of a polynomial function 2.To use the Rational Zero Theorem to find the possible rational zeros.

Find Rational Zeros, I Objectives: 1.To find the zeros of a polynomial function 2.To use the Rational Zero Theorem to find the possible rational zeros.

ON EXTENSIONS OF RATIONAL MODULESON EXTENSIONS OF RATIONAL MODULES 3 Theorem A coalgebra has a left rational torsion functor if and only if the open co nite ideals are closed under

ON EXTENSIONS OF RATIONAL MODULESON EXTENSIONS OF RATIONAL MODULES 3 Theorem A coalgebra has a left rational torsion functor if and only if the open co nite ideals are closed under

The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives.

The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives.

Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.

Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem Represent a polynomial equation of degree n. If a rational number, where p and q have.

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem Represent a polynomial equation of degree n. If a rational number, where p and q have.

3.3 - 1 3.3 Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.

3.3 - 1 3.3 Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.

Chapter 2: Polynomial and Rational Functions 2b... · Finding Actual Zeros of a Polynomial using the Rational Zero Theorem Once we find a list of all possible rational zeros, we will

Chapter 2: Polynomial and Rational Functions 2b... · Finding Actual Zeros of a Polynomial using the Rational Zero Theorem Once we find a list of all possible rational zeros, we will

Objectives Identify the multiplicity of roots Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations.

Objectives Identify the multiplicity of roots Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations.

Acquisition Lesson Planning Form - ciclt. · PDF fileAcquisition Lesson Planning Form Plan for the Concept, ... remainder theorem, rational root theorem, ... incorporating complex

Acquisition Lesson Planning Form - ciclt. · PDF fileAcquisition Lesson Planning Form Plan for the Concept, ... remainder theorem, rational root theorem, ... incorporating complex

Agenda – Jan 6 Do Now Review Reading Notes: Rational Root Theorem Work Time Due NEXT CLASS: Rational Root Theorem By the end of today’s class,

Agenda – Jan 6 Do Now Review Reading Notes: Rational Root Theorem Work Time Due NEXT CLASS: Rational Root Theorem By the end of today’s class,

Confluence of singular fibers on rational elliptic surfaces · Confluence of singular fibers on rational elliptic surfaces Results Theorem Of all confluences to Singular Fibers

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Bell Ringer. Rational Root Theorem November 4, 2015.

Bell Ringer. Rational Root Theorem November 4, 2015.

The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine.

The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine.