POLYNOMIALS

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POLYNOMIALS

• CHAPTER 3
• DCT1043

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CONTENT

• 3.1 Polynomial Identities
• 3.2 Remainder Theorem,
• Factor Theorem and
• Zeros of Polynomial
• 3.3 Partial Fractions Decomposition

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3.1POLYNOMIALSIDENTITIES
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Objectives

• By the end of this topic, you should be able to
• Recognize monomials, binomials trinomials
• Define polynomials, state the degree of a
  polynomial the leading coefficient
• Perform addition, subtraction multiplication of
  polynomials
• Perform division of polynomials

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Introduction Basic Vocabulary

• Monomial
• where a is a constant (coefficient) and
  the power n is a non-negative integer called
  degree
• Nonmonomial
• Binomial
• The sum or difference of 2 monomial having
  different degrees
• Trinomial
• The sum or difference of 3 monomial having
  different degrees
• Polynomial
• The sum of monomials

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Polynomial Identities

• Polynomials
• A word derived from Greek many terms
• Polynomial in the variable x in algebraic
  expression
• Polynomial is write in the form
• where is a constants
  (coefficients), is the leading
• coefficient and the power n is a non-negative
  integer called the
• degree of the polynomial (The highest power of x)
• Identity
• 2 equation which have the same solution though
  expressed differently ( use (equivalent)
  sign)

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Example1 (Identity)

• Given that
  for all values of x.
  Find the value of a, b
  and c.
• Given that
  for all values of x.
  Find the value of a
  and b.

TIPS To find unknowns in an identity, a)
Substitute suitable values of x, or b) Equate
coefficients of like powers of x
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Adding Subtracting Polynomials

• Horizontal Addition Subtraction
• Group the like terms (monomials with the same
  power) and then combine them
• Vertical Addition Subtraction
• Vertically line up the like terms in each
  polynomial and then add or subtract the
  coefficients.

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Example 2 (Adding Subtracting)
A B C D E
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Multiplying Polynomials

• Horizontal Multiplication
• Using the Laws of exponents, the Commutative,
  Associative and Distributive properties and then
  combine them
• Vertical Multiplication
• Write the polynomial with the greatest number of
  terms in the top row.

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Example 3 (Horizontal Multiplication)
A B C D E
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Example 4 (Vertical Multiplication)
A B C
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Special Product Formulas
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Example 5 (Special Product)
A B C D E
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Factoring Polynomials

• Any polynomial
• Look for common monomial factors
• Binomials of degree 2 or higher
• Check for a special products
• Trinomials of degree 2
• Check for a perfect square
• Three or more terms
• Grouping factored out the common factor from
  each of several groups of terms.

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Factoring Strategy

• Factor out the Greatest Common Factors (GCF)
• Check for any Special Products
• 2 term or 3 term
• If not a perfect square use try and error or
  grouping
• See if any factors can be factored further

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Example 6 (Factoring any polynomial)
A B C D E
F G H I J
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Example 7 (Factoring special forms)
A B C D E
F G H I J
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Division of a Polynomial

• RECALL Dividing Two integers

• Long Division for polynomials
• The process is similar like division for integers
• The process is stop when the degree of the
  remainder is less than the degree of divisor

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Example 8 (Long Division)
A B C D E
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3.2REMAINDER THEOREM, FACTOR THEOREM ZEROS
OF POLYNOMIALS
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Objectives

• At the end of this topic, you should be able to
• Use the remainder and factor theorem
• Identify the value of a such that (x a) is a
  factor of P (x) and factorize P (x) completely
• Find the roots and the zeros of a polynomial
• Determine the complex zeros of a polynomial up to
  degree three

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Remainder Theorem

• The remainder theorem state that,

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Example 9 (Remainder Theorem)
Find the remainder if is
divided by
A B C D
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Example 10 (Remainder Theorem)
(a) The expression leaves
a remainder -2 when divided by
Find the value of p
(b) Given that the expression
leaves the same remainder
when divided by or by .
Prove that
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Factor Theorem

• The factor theorem state that,
• Means that,

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Example 11 (Factor Theorem)
(a) Determine whether or not is a
factor of the following polynomials.
i) ii)
(b) Determine whether or not and
is a factor of
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Zeros of Polynomial

• Use to solve polynomial function
• A zero (root) of a function f is any value of x
  for which f (x) 0
• Number of real zeros
• A polynomial function cannot have more real zeros
  than its degree
• The maximum number are n

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Descartes Rule of Signs

• Let f denote a polynomial function written in
  standard form.
• The number of positive real zeros of f either
  equals the number of variation in the sign of the
  nonzero coefficients of f (x) or else equals that
  number less an even integer (2)
• The number of negative real zeros of f either
  equals the number of variation in the sign of the
  nonzero coefficients of f (-x) or else equals
  that number less an even integer (2)

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Example 12 (Descartes Rule of Signs)
Determine the number of maximum real zeros,
positive real zeros and negative real zeros from
the following polynomials.
i) ii)
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Rational Zeros Theorem

• Let f be a polynomial function of degree 1 or
  higher of the form
• where each coefficient is an integer.
• If in lowest terms, is a rational
  zero of f, then p must be a factor of and
  q must be a factor of

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Example 13 (Rational Zeros Theorem)
Listing all the potential real zeros from the
following polynomials.
i) ii)
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Finding the Real Zeros

• Step 1 Determine the maximum number of zeros
  degree
• Step 2 Determine the number of positive
  negative zeros
• Descartes Rule of Signs
• Step 3 Identify those rational numbers that
  potentially can
• be zeros Rational Zeros
  Theorem
• Step 4 Test each potential rational zeros long
  division
• Step 5 Repeat Step 3 if a zero is found
• Step 6 If possible, use the factoring techniques
  to find the
• zeros

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Example 14 (Finding Real Zeros)
Find all the real zeros from the following
polynomials.
i) ii)
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3.3PARTIAL FRACTION
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Objectives

• By the end of this topic, you should be able to
• Define partial fractions
• Obtain partial fractions decomposition when the
  denominators are in the form of
• A linear factor
• A repeated linear factor
• A Quadratic factor that cannot be factorized
• A repeated quadratic factor

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What is partial fractions

• Consider the problem of adding 2 fraction
• The reverse procedure

Partial fraction
Partial fraction decomposition
Partial fraction
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What is partial fractions

• Any rational function
• where the degree of P is less than the degree
  of Q, could be expressed as a sum of relatively
  simpler rational functions, called partial
  fractions.
• If f (x) is improper (degree of Q is less than
  the degree of P), then by long division, dividing
  P by Q until a remainder R (x) is obtained such
  that degree of R is less than the degree of Q.

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Case 1 The denominator Q (x) is a product of
distinct linear factor

• Examples

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Case 2 The denominator Q (x) is a product of
repeated linear factors

• Examples

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Case 3 The denominator Q (x) contains
irreducible quadratic factors

• Examples

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Case 4 Q (x) contains a repeated irreducible
quadratic factors

• Examples

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