If f(x) is a polynomial and f(a) = 0, then (x–a) is a factor of f(x) Proof of the factor theorem Let's start with an example Consider 4 8 5
The Factor Theorem: Suppose p is a nonzero polynomial It is important to note that it works only for these kinds of divisors 5 Also Example 3 2 1
Now consider another example of a cubic polynomial divided by a linear divisor From the above example, we can deduce that: 2 ? 3 + 4 + 5 =
1 Factorise polynomial expressions 2 3 2 Divide a polynomial by a linear or quadratic factor 2 3 3 Apply the remainder theorem 2 3 4 Apply the factor theorem
For example, the zeros of p are –3, 1, and 5, and the factors of p(x) are x + 3, x - 1, and x - 5 The following theorem generalizes this relationship
2 Remainder and Factor Theorems Interactive Mathematics Factor theorem state with proof examples and solutions factorise the Polynomials Maths Mutt
Example 8: 7 5 4 3)( 2 3 + ? + = x x x xf Find )4( ? f using (a) synthetic division (b) the Remainder Theorem Example 9: Solve the equation
4 2 - Algebra - Solving Equations 4 2 8 - The Factor Theorem Higher Level ONLY 1 / 5 Example 1 Q Suppose f (x)=5x3 - 14x2 + 12x - 3
24 fév 2015 · Use long division to determine the other factors Page 6 6 February 24, 2015 Example Five Factor fully
PROBLEM You Have the Right to the Remainder Theorem Chapter 5 Polynomial Expressions and Equations example are correct? - Long Division
101353_62_3_factor_and_remainder_theorems.pdf
2.3 Factor and remainder theorems
Dr Richard Harrison
FEPS Mathematics Support Framework
Core
Preparatory
Topics
1.1 1.2 2.1 2.2 2.3 3.1 5.1 5.2 5.3 9.1 10.1 11.1 11.5
2.3 Introduction
The aim of this unit is to assist you in consolidating and developing your knowledge and skills in working with the factor and remainder theorems. It will also refresh your skills in algebraic manipulation and in solving two linear equations simultaneously.
While studying
the slides. After studying the slides, you should attempt the Consolidation Questions. #uniofsurrey2
2.3 Learning checklist
#uniofsurrey3
Learning
resource
NotesTick when
complete
Slides
Your turn
questions
Consolidation
questions
2.3 Learning objectives
After completing this unit you should be able to
2.3.1 Factorise polynomial expressions
2.3.2 Divide a polynomial by a linear or quadratic factor
2.3.3 Apply the remainder theorem
2.3.4 Apply the factor theorem
#uniofsurrey4
Some terminology (1)
Identity: a statement that two mathematical expressions are equal for all values of their variables (symbol ). Irreducible quadratic: a quadratic expression which cannot be written as a product of two linear factors using the set of real numbers, R, e.g., ݔʹͳൌͲ. In other words, it is a quadratic with a negative discriminant (ܾʹȂͶܿܽ Linear: a function or expression containing a variable with a first degree power, e.g., ݔͳǡʹݔݕ.
Some terminology (2)
Polynomial: the general form of a polynomial expression is the sum of terms, ܽ݊ݔ݊ܽ݊െͳݔ݊െͳǥܽ
The highest power, n is the degree or order of the polynomial (it is also the dominant term). The powers in each term are non-negative. ܽ݊is called the leading coefficient and ܽ
Quotient: the result of dividing one number (or polynomial) by another.
2.3.1. Factorise polynomial
expressions Factorisationis the process of rewriting an algebraic (often a polynomial) expression as a product of simpler, irreduciblefactors. The original expression is divisible by its factors. Eg1, a quadratic expression can sometimes be expressed as a product of two linear factors:
ݔʹȂʹݔȂؠ
Eg2, an irreducible quadratic is a quadratic expression that cannot be factorised, x2+ 1, for instance.
ݔ͵ݔʹݔͳؠ
Cannot be simplified further in the real number system Note
Any polynomial with real coefficients can be written as a product of linear factors and irreducible quadratic factors.
Most polynomials that arise in real world applications cannot be factorisedfactorisebut to find the roots of the polynomial. This is done numerically with software.
Clearly, if we can factorisethe polynomial manually, we can also write down the roots. For now, we are concerned only with real roots.
There are general algebraic solutions to cubic and quartic polynomial equations (analogous to the quadratic formula).
Some useful identities
Using the greatest common factor
(GCF) to factorise
Factorisingby grouping
Factorisationusing identities
Using a substitution to factorise
Factorisationsummary
In general, only relatively simple expressions can be factorised easily. We can sometimes use GCF, grouping, identities or substitutions to help.
Trial and error may be involved!
There are other useful techniques available to us for finding any factors of polynomial expressions and solving polynomial equations. These will be considered next.
Your turn! (1)
Factorise the following,
ݔସെʹͷ
ݔସെ͵ݔଷെݔ͵ #universityofsurrey15
Solutions
ݔସെʹͷൌݔଶʹͷݔଶെʹͷ
ݔସെ͵ݔଷെݔ͵ൌݔସെݔെ͵ݔଷ͵
ൌݔݔଷെͳെ͵ሺݔଷെͳሻ ൌݔെ͵ሺݔଷെͳሻ #universityofsurrey16
2.3.2. Divide a polynomial by a linear or
quadratic factor
How can we use this to solve
equations?
Example long division
Example: long division
The quotient factorises so we can write the
polynomial as a product of three linear factors
Example: Long division
ሺݔȂͳሻଶ
Your turn! (2)
Divide ݔଷെ͵ݔଶͶݔെͳʹܾ #universityofsurrey22
Solution
Quotient = ݔଶͶ
Remainder = 0
#universityofsurrey23
2.3.3. Apply the remainder theorem
The remainder theorem
Example: Remainder theorem
Example: Remainder theorem
Your turn! (3)
Find the remainder when ݔଷെͷݔଶെʹݔെͷis divided by ሺݔ͵ሻ
#universityofsurrey28
Solution
݂െ͵ൌሺെ͵ሻଷെͷെ͵ଶെʹെ͵െͷൌെͳ
#universityofsurrey29
2.3.4. Apply the factor theorem
Why the factor theorem is useful
We can use the factor theorem to help us
factorisepolynomials and to solve polynomial equations!
Knowing ሺݔെܽ
also know ܽ
Example: Factor theorem
Example: Factor theorem
Your turn! (4)
Find all the solutions to ݔଷݔଶെ͵ݔ͵ͷ= 0 #universityofsurrey34
Solution
Try ݂ሺͳሻfirst
݂ͳൌͳͳെ͵͵ͷൌͲฺሺݔെͳሻis a factor
Now try ݂ሺͷሻ
݂ͷൌͳʹͷʹͷെͳͺͷ͵ͷൌͲฺሺݔെͷሻis a factor
Finally try ݂ሺെሻ
݂ͷൌെ͵Ͷ͵Ͷͻʹͷͻ͵ͷൌͲฺሺݔሻis a factor
Roots: ݔא
#universityofsurrey35
Further examples: Factor and
remainder theorem
Further examples: Factor and
remainder theorem
Further examples: Factor and
remainder theorem
Further examples: Factor and
remainder theorem
2.3 Summary
You should now be able to,
2.3.1Factorise polynomial expressions
2.3.2Divide a polynomial by a linear or quadratic factor
2.3.3Apply the remainder theorem
2.3.4Apply the factor theorem
#uniofsurrey40 41