Maths Extension 2 - Polynomials. Polynomials
|
|
- Alexandra Powell
- 5 years ago
- Views:
Transcription
1 Maths Extension - Polynomials Polynomials! Definitions and properties of polynomials! Factors & Roots! Fields ~ Q Rational ~ R Real ~ C Complex! Finding zeros over the complex field! Factorization & Division of polynomials! Remainder & Factor Theorem! Rational Roots! Multiplicity Theorem/ Repeated Roots! Relationship between the roots and coefficients of a polynomial equation
2 Maths Extension - Polynomials Definitions and properties of polynomials Polynomial Expression P(x) = p x n + p x n- + p x n- + + p n- x + p n where p Coefficients Leading term p, p, p, p, p n x n Constant p If p n = If p = p = p = It is a monic Then P(x) is a zero polynomial P(x) = x x + 7x x + Coefficient of.x Is.x Is Leading term is x Constant is More points: _a i _a _a n x n _a n _P(x) _a n = _P(x) _P(x) _P(x) = Are the coefficients of the polynomial Is the constant term Is the leading term Is the leading coefficient Polynomial of degree n The polynomial is a monic Null polynomial Expression of polynomial Polynomial equation
3 Maths Extension - Polynomials Factors and Roots Factor A polynomial that divides into another and has remainder (x ) is a factor of x Root P() is a root of x _x = x = Fields Q Rational R Real Integer numbers ±,,, Irrational Numbers Surdic roots occur in conjugate pairs If a + b is a root, so too will a b C Complex Numbers over the complex field a ± ib Complex roots occur in conjugate pairs If a + ib is a root, so too will a ib Factorize x x 5 over Q, R, C Y Y - 5 = (Y 5)(Y + ) Q = (x 5)(x + ) R = (x + 5 )(x 5 )(x + ) C = (x + 5 )(x 5 )(x + i )(x ) i
4 Maths Extension - Polynomials Finding zeros over the complex field. If one root is complex, then one of the other roots is it s conjugate. If ( + i ) is a root, [x ( + i )] is the factor So [x ( + i )][x ( i )] = x x + How do we get this??? (x z)(x - z ) Where: _z = + i = x (z + z )x + z z z = - i _z + z = = x x + z z = Find all the zeros of P(x) = x x x + 6x over C. ( + i ) is a zero. If ( + i ) is a root, then ( i ) is also a root. By multiplying out the factors: x x + _x + x _x x + _x x x + 6x _x x +x _x x + 6x _x x + x x + x x + x The factorized equation is [x ( + i )][x ( i )](x + )(x ) The zeros are ( + i ), ( i ), -,
5 Maths Extension - Polynomials Factorization and Division of Polynomials Factorizing polynomials. Simple factorizing. Trinomials, Grouping, Difference of squares, etc. Quadratic Formula. Completing the Square Simple factorizing _x + x = (x + )(x ) _x = (x )(x + ) = (x )(x + )(x + ) = (x )(x + )(x + i )(x - i ) Quadratic formula _x + x + _x = ± () ()() ± 8 = ± = = ± i (x a) _x + x + _x = (x + + i )(x + ) i ± () ()() ± = 8 ± i = 8 i i ( x + + )( x + ) We can only find the factors of the polynomial, not the constant outside Completing the Square _x + x + = x + x + + = (x + ) + = (x + ) i = [(x + ) i ][(x + ) + ] i = (x + i )(x + + ) i x + + _x + x + = [ ] 9 = [( x + + ) + ] 6 6 i [ x + ] 8 6 = ( ) i i = [ x + + ][ x + ] ! Complete the square, then add end term to satisfy the equation.! i = -! Difference of two squares 5
6 Maths Extension - Polynomials Division of polynomials P(x) = A(x) Q(x) + R(x) Dividend = Divisor Quotient + Remainder x x + 7x x + = x x + 5x +7x LONG DIVISION!!! x + 5x + 7x + x x x + 7x x + x 6x 5x + 7x 5x x 7x 7x Divide and find a such that R(x) = x x x + x 6 67 x + (a )x + ( a) x + x + ax + ax + 6 x + ax (a )x + ax (a )x + (z )x ( a)x + 6 ( a)x + ( a) a For R(x) =, a a = = a = 6
7 Maths Extension - Polynomials Factor and Remainder Theorems Remainder Theorem! If a polynomial P(x) is divided by (x a), then the remainder is P(a) x ; a = P() = () () + 7() () + = = 67 x + 5x + 7x + x x x + 7x x + x 6x 5x + 7x 5x x 7x 7x x x x + x 6 67 Factor Theorem! For any polynomial P(x), if P(a) =, then (x a) is a factor of P(x) OR! For any polynomial P(x), if (x a) is a factor of P(x), then P(a) = 7
8 Maths Extension - Polynomials Rational Roots Let P(x) have degree n with integer coefficients. Suppose P(x) has a rational root of p q. _p, q, are prime integers. Then p a and q a n is divides into Given that P(x) = x x x + 6 has a rational root. Find all the zeros of P(x). Let the root be p q. _p 6_;_q The possible rational roots are: _p ± 6, ±, ±, ± _q ±, ± P(-) = P() = P( ) = The zeros of P(x) are,, The factorized equation is (x + )(x )(x - ) OR (x + )(x )(x ) Given that P(x) = x + x + x 6 has a rational root. Find all the zeros of P(x). Let the root be p q. _p -6_;_q The possible rational roots are: _p ± 6, ±, ±, ± _q ± P() = (x ) is a factor _x + 5x + 6 _x _x + x + x 6 _x x (x )(x + )(x + ) 5x + x 5x 5x 6x 6 6x 6 The zeros of P(x) are, -, - The factorized equation is (x + )(x + )(x ) 8
9 Maths Extension - Polynomials Multiplicity Repeated Roots _x = a is a repeated root of P(x) or multiplicity r. If P(x) = (x a) r.q(x) so (x a) / Q(x) If x = a is of multiplicity r of P(x), then multiplicity (r ) or P`(x) Proof Let P(x) = (x a) r.q(x) P`(x) = (x a) r.q`(x) + Q(x).r(x a) r-. Product rule = (x a) r- [(x a).q(x) + r.q(x)] = (x a) r-.r(x) P`(x) has a root at x = a with multiplicity (r ) Find roots of P(x) = x + x, given that the polynomial has a double root. P(x) = x + x P`(x) _x = - is a double root. = x + 6x P(x) has a single root, so only need to differentiate once. = x(x + ) _x = or Check both results by remainder theorem!!! P() P(-) = To find the other root, we can either use Long Division or Sum of the Roots Long Division (x + ) = x + x + _x x + x + _x + x + _x + x + x x x x x Sum of the Roots Sum of the roots one at a time = b a α + β + γ = + ( ) + γ = γ = So the roots of P(x) are: -, -, 9
10 Maths Extension - Polynomials For polynomials which have: Double roots Differentiate one time Triple roots Differentiate two times Quadruple roots Differentiate three times If number of multiple roots are not given, keep differentiating until the P`(x), P``(x), etc.. can be factorized. Factorize and find the zeros of P(x) = x + x x 5x if it has multiple roots. P(x) = x + x x 5x P`(x) = x + x 6x 5 P``(x) = x + 6x 6 = 6(x )(x + ) P(-) = α + β + γ + δ + δ δ = = = P(x) = x x 9x + C has a double root. Find C P`(x) = x 6x 9 = (x )(x + ) _x = or We can substitute both, but our C will be different. Both answers must be given P() = C C = 7 P(-) = C C = 5
11 Maths Extension - Polynomials Relationship between the roots and coefficients of a polynomial equation Quadratic : ax + bx + c α + β b = Sum of roots at a time a αβ c = a Sum of roots at a time (product of roots) Cubic : ax + bx + cx + d α + β + γ b = Sum of roots at a time a αβ + βγ + c γα = a Sum of roots at a time αβγ d = a Sum of roots at a time (product of roots) Quartic : ax + bx + cx + dx + e α + β + γ + δ b = Sum of roots at a time a αβ + βγ + γδ + δα + αγ + βδ c = a Sum of roots at a time αβγ + βγδ + γδα + δαβ d = Sum of roots at a time a αβγδ e = a Sum of roots at a time (product of roots) If n is the number of roots at a time, to find out how many combinations there are, we use: n C r. For a quartic: α + β + γ + δ = Σ α = Σα i C αβ + βγ + γδ + δα + αγ + βδ = Σ αβ = Σα i α j C αβγ + βγδ + γδα + δαβ = Σ αβγ = Σαiα jα k C αβγδ = Σ αβγδ = Σαiα jα kα l C
12 Maths Extension - Polynomials Roots are α, β,γ for x x x. ( α )( β )( γ ) = αβγ ( αβ + βγ + γα) + ( α + β + γ ) = + =. ( β + γ α)( γ + α β )( α + β γ ) If α + β + γ = α + β γ = γ β + γ α = α γ + α β = β = ( α )( β )( γ ) = 8( α )( β )( γ ) From part = 8 =. Σα i = ( Σαi ) ( Σαiα j ) = - ( ) = + = 7. Σα i α β γ α β γ α β γ = = = α i ( Σ ) ( Σα i ) ( Σαi) = αi Σ 8 6 = αi Σ = 7 7 Σ α i =
13 Maths Extension - Polynomials 5. Σα i P(x) is a cubic x.p(x) is now a quadratic Sub it in x x x x x x x α β γ α β γ α β γ α α α 6. Σα β Σ = α i α j α i ( Σ ) ( Σα i ) ( Σα i ) αi 7 Σ ( ) αi Σαi Σαi (7) Σ 7 97 = α β + β γ + γ α = ( Σα α ) ( Σα α α )( Σα) i j = ) ( )() ( = i j k 7. Σ α α β γ = + + α i j = ( ) = i α α α α = j k 8. α β α Σ i α j = α β + α γ + β α + β γ + γ α + γ β = ( α + β + γ )( αβ + βγ + γα) αβγ = ( Σαi)( Σαiα j ) ( Σαiα jα k ) = )( ) ( ) ( 9 =
14 Maths Extension - Polynomials Relationship & Transformation methods If α, β,γ are the roots of x 5x + x + 6 =, form the equation whose roots are: α, β, γ Relationship Method: Sum of the roots one at a time α + β + γ = ( α + β + γ ) 5 = () = 5 Sum of the roots two at a time ( α )(β ) + (β )(γ ) + (γ )(α ) = ( αβ + βγ + γα) = () = 8 Sum of the roots three at a time ( α )(β )(γ ) = 8( αβγ ) = 8( ) = The new equation is x 5x + 8x + = Transformation Method: x = α, β, γ _y = x y = α,β, γ y _x = = y y y = y y y = y 5y y + 6 = y 5y 8y Change y for x = x 5x 8x The Transformation Method is preferred
15 Maths Extension - Polynomials If α, β,γ are the roots of x + x + x 5 =, form the equation whose roots are: A),, B) α, β, γ C) + α, + β, + γ α β γ D) α, β, γ E) α, β, A) x + x + x 5 = x = y B) x + x + x 5 = x = y + y y y + + y y y = () () + 5 = 5 = + y + y 5y = 5x x x y y y = () + () + 5 y y y = y + y = y = x + x + x γ F) α, C) x + x + x 5 = x = y = (y ) + (y ) + (y ) 5 = (y )(y )(y ) + (y )(y ) + (y ) 5 = (y 6y + y 8) + (y y + ) + (y ) 5 = y 9y + y = x 9x + x β, γ, D) x + x + x 5 = x = y + = (y + ) + (y + ) + (y + ) 5 = (y + )(y + )(y + ) + (y + )(y + ) + (y + ) 5 = (y + 6y + y + 8) + (y + y + ) + (y + ) 5 = y + 5y + 7y + 5 = x + 5x + 7x
16 Maths Extension - Polynomials E) x + x + x 5 = x = y = y + y + y 5 = y + y + y 5 y 5 = y ( y + ) Square both sides 9y y + 5 = y(y + y + ) = y + y + y = y 5y + y 5 = x 5x + x 5 F) x + x + x 5 = x = y = (-y) + (-y) + (-y) 5 = y + y y 5 = y y + y + 5 = x x + x
4 Unit Math Homework for Year 12
Yimin Math Centre 4 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 3 Topic 3 Polynomials Part 2 1 3.2 Factorisation of polynomials and fundamental theorem of algebra...........
More informationQuestion 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.
Class X - NCERT Maths EXERCISE NO:.1 Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) (iv) (v)
More informationB.Sc. MATHEMATICS I YEAR
B.Sc. MATHEMATICS I YEAR DJMB : ALGEBRA AND SEQUENCES AND SERIES SYLLABUS Unit I: Theory of equation: Every equation f(x) = 0 of n th degree has n roots, Symmetric functions of the roots in terms of the
More informationTenth Maths Polynomials
Tenth Maths Polynomials Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non-negative integer
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationPolynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:
Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More information(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =
Dividing Polynomials Prepared by: Sa diyya Hendrickson Name: Date: Let s begin by recalling the process of long division for numbers. Consider the following fraction: Recall that fractions are just division
More informationMore Polynomial Equations Section 6.4
MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division
More informationOCR Maths FP1. Topic Questions from Papers. Roots of Polynomial Equations
OCR Maths FP1 Topic Questions from Papers Roots of Polynomial Equations PhysicsAndMathsTutor.com 18 (a) The quadratic equation x 2 2x + 4 = 0hasroots and. (i) Write down the values of + and. [2] (ii) Show
More informationDividing Polynomials: Remainder and Factor Theorems
Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.
More information, a 1. , a 2. ,..., a n
CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.
More informationPolynomials. Henry Liu, 25 November 2004
Introduction Polynomials Henry Liu, 25 November 2004 henryliu@memphis.edu This brief set of notes contains some basic ideas and the most well-known theorems about polynomials. I have not gone into deep
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More informationNotes on Polynomials from Barry Monson, UNB
Notes on Polynomials from Barry Monson, UNB 1. Here are some polynomials and their degrees: polynomial degree note 6x 4 8x 3 +21x 2 +7x 2 4 quartic 2x 3 +0x 2 + 3x + 2 3 cubic 2 2x 3 + 3x + 2 3 the same
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationPOLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1
POLYNOMIALS A polynomial in x is an expression of the form p(x) = a 0 + a 1 x + a x +. + a n x n Where a 0, a 1, a. a n are real numbers and n is a non-negative integer and a n 0. A polynomial having only
More informationUNIT-I CURVE FITTING AND THEORY OF EQUATIONS
Part-A 1. Define linear law. The relation between the variables x & y is liner. Let y = ax + b (1) If the points (x i, y i ) are plotted in the graph sheet, they should lie on a straight line. a is the
More informationWarm-Up. Use long division to divide 5 into
Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x
More informationMEMORIAL UNIVERSITY OF NEWFOUNDLAND
MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS Section 5. Math 090 Fall 009 SOLUTIONS. a) Using long division of polynomials, we have x + x x x + ) x 4 4x + x + 0x x 4 6x
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that
More informationDraft Version 1 Mark scheme Further Maths Core Pure (AS/Year 1) Unit Test 5: Algebra and Functions. Q Scheme Marks AOs. Notes
1 b Uses α + β = to write 4p = 6 a TBC Solves to find 3 p = Uses c αβ = to write a 30 3p = k Solves to find k = 40 9 (4) (4 marks) Education Ltd 018. Copying permitted for purchasing institution only.
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More informationPolynomial Functions and Models
1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 = + x 3 x +. The point is that we don
More informationClass IX Chapter 2 Polynomials Maths
NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the
More information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
More informationx 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
More informationSection 3.1: Characteristics of Polynomial Functions
Chapter 3: Polynomial Functions Section 3.1: Characteristics of Polynomial Functions pg 107 Polynomial Function: a function of the form f(x) = a n x n + a n 1 x n 1 +a n 2 x n 2 +...+a 2 x 2 +a 1 x+a 0
More informationHow might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5
8.4 1 8.4 Partial Fractions Consider the following integral. 13 2x (1) x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that 13 2x (2) x 2 x 2 = 3 x 2 5 x + 1 We could then
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
More informationAssessment Exemplars: Polynomials, Radical and Rational Functions & Equations
Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More information(x + 1)(x 2) = 4. x
dvanced Integration Techniques: Partial Fractions The method of partial fractions can occasionally make it possible to find the integral of a quotient of rational functions. Partial fractions gives us
More information2 the maximum/minimum value is ( ).
Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationMTH310 EXAM 2 REVIEW
MTH310 EXAM 2 REVIEW SA LI 4.1 Polynomial Arithmetic and the Division Algorithm A. Polynomial Arithmetic *Polynomial Rings If R is a ring, then there exists a ring T containing an element x that is not
More informationPartial Fractions. Calculus 2 Lia Vas
Calculus Lia Vas Partial Fractions rational function is a quotient of two polynomial functions The method of partial fractions is a general method for evaluating integrals of rational function The idea
More informationPolynomial expression
1 Polynomial expression Polynomial expression A expression S(x) in one variable x is an algebraic expression in x term as Where an,an-1,,a,a0 are constant and real numbers and an is not equal to zero Some
More informationP4 Polynomials and P5 Factoring Polynomials
P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationChapter 2.7 and 7.3. Lecture 5
Chapter 2.7 and 7.3 Chapter 2 Polynomial and Rational Functions 2.1 Complex Numbers 2.2 Quadratic Functions 2.3 Polynomial Functions and Their Graphs 2.4 Dividing Polynomials; Remainder and Factor Theorems
More information3 What is the degree of the polynomial function that generates the data shown below?
hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior
More informationMath From Scratch Lesson 37: Roots of Cubic Equations
Math From Scratch Lesson 7: Roots of Cubic Equations W. Blaine Dowler September 1, 201 Contents 1 Defining Cubic Equations 1 2 The Roots of Cubic Equations 1 2.1 Case 1: a 2 = a 1 = 0.........................
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include
PUTNAM TRAINING POLYNOMIALS (Last updated: December 11, 2017) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationUnit 2 Rational Functionals Exercises MHF 4UI Page 1
Unit 2 Rational Functionals Exercises MHF 4UI Page Exercises 2.: Division of Polynomials. Divide, assuming the divisor is not equal to zero. a) x 3 + 2x 2 7x + 4 ) x + ) b) 3x 4 4x 2 2x + 3 ) x 4) 7. *)
More informationa real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials
5-1 Polynomial Functions Objectives A2.A.APR.A.2 (formerly A-APR.A.3) Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
More informationMath 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...
Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form... where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationReview all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationAS1051: Mathematics. 0. Introduction
AS1051: Mathematics 0 Introduction The aim of this course is to review the basic mathematics which you have already learnt during A-level, and then develop it further You should find it almost entirely
More informationToday. Polynomials. Secret Sharing.
Today. Polynomials. Secret Sharing. A secret! I have a secret! A number from 0 to 10. What is it? Any one of you knows nothing! Any two of you can figure it out! Example Applications: Nuclear launch: need
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More information2.1. The Remainder Theorem. How do you divide using long division?
.1 The Remainder Theorem A manufacturer of cardboard boxes receives an order for gift boxes. Based on cost calculations, the volume, V, of each box to be constructed can be modelled by the polynomial function
More informationOperations w/polynomials 4.0 Class:
Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the y-intercept, the equation
More informationUnit 1: Polynomial Functions SuggestedTime:14 hours
Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an
More information7.4: Integration of rational functions
A rational function is a function of the form: f (x) = P(x) Q(x), where P(x) and Q(x) are polynomials in x. P(x) = a n x n + a n 1 x n 1 + + a 0. Q(x) = b m x m + b m 1 x m 1 + + b 0. How to express a
More informationFunctions and Equations
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN
More informationPolynomials. Chapter 4
Chapter 4 Polynomials In this Chapter we shall see that everything we did with integers in the last Chapter we can also do with polynomials. Fix a field F (e.g. F = Q, R, C or Z/(p) for a prime p). Notation
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by MATH 1700 MATH 1700 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 2 / 11 Rational functions A rational function is one of the form where P and Q are
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationIntegration of Rational Functions by Partial Fractions
Title Integration of Rational Functions by Partial Fractions MATH 1700 December 6, 2016 MATH 1700 Partial Fractions December 6, 2016 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 Partial Fractions
More informationChapter 4. Remember: F will always stand for a field.
Chapter 4 Remember: F will always stand for a field. 4.1 10. Take f(x) = x F [x]. Could there be a polynomial g(x) F [x] such that f(x)g(x) = 1 F? Could f(x) be a unit? 19. Compare with Problem #21(c).
More informationMathematical Olympiad Training Polynomials
Mathematical Olympiad Training Polynomials Definition A polynomial over a ring R(Z, Q, R, C) in x is an expression of the form p(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, a i R, for 0 i n. If a n 0,
More informationHonours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period:
Honours Advanced Algebra Name: Unit : Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Mathematical Goals Know and apply the Remainder Theorem Know and apply
More information7x 5 x 2 x + 2. = 7x 5. (x + 1)(x 2). 4 x
Advanced Integration Techniques: Partial Fractions The method of partial fractions can occasionally make it possible to find the integral of a quotient of rational functions. Partial fractions gives us
More informationSection 4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem
Section 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem Polynomial Long Division Let s compute 823 5 : Example of Long Division of Numbers Example of Long Division of Numbers Let
More information8.4 Partial Fractions
8.4 1 8.4 Partial Fractions Consider the following integral. (1) 13 2x x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that (2) 13 2x x 2 x 2 = 3 x 2 5 x+1 We could then
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationSimplifying Rational Expressions and Functions
Department of Mathematics Grossmont College October 15, 2012 Recall: The Number Types Definition The set of whole numbers, ={0, 1, 2, 3, 4,...} is the set of natural numbers unioned with zero, written
More informationRings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.
Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary
More information1) Synthetic Division: The Process. (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem
J.F. Antona 1 Maths Dep. I.E.S. Jovellanos 1) Synthetic Division: The Process (Ruffini's rule) 2) Remainder Theorem 3) Factor Theorem 1) Synthetic division. Ruffini s rule Synthetic division (Ruffini s
More informationMULTIPLYING TRINOMIALS
Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationPOLYNOMIALS. Maths 4 th ESO José Jaime Noguera
POLYNOMIALS Maths 4 th ESO José Jaime Noguera 1 Algebraic expressions Book, page 26 YOUR TURN: exercises 1, 2, 3. Exercise: Find the numerical value of the algebraic expression xy 2 8x + y, knowing that
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2
More informationPower and Polynomial Functions. College Algebra
Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
More informationMath123 Lecture 1. Dr. Robert C. Busby. Lecturer: Office: Korman 266 Phone :
Lecturer: Math1 Lecture 1 Dr. Robert C. Busby Office: Korman 66 Phone : 15-895-1957 Email: rbusby@mcs.drexel.edu Course Web Site: http://www.mcs.drexel.edu/classes/calculus/math1_spring0/ (Links are case
More information0. Introduction. Math 407: Modern Algebra I. Robert Campbell. January 29, 2008 UMBC. Robert Campbell (UMBC) 0. Introduction January 29, / 22
0. Introduction Math 407: Modern Algebra I Robert Campbell UMBC January 29, 2008 Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22 Outline 1 Math 407: Abstract Algebra 2 Sources 3 Cast of
More informationMATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS
E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 MATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS dt dt Key knowledge The key features and properties of cubic polynomials functions and
More information