How to find zeros or factor polynomials of high degree

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Chapter 5

• How to find zeros or factor polynomials of high
  degree

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Polynomials to consider

• We consider polynomials whose coefficients are
• Real numbers
• integers

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Factor Theorem

• The relation between zeros and factorization
• Examples

1. If f(c)0, then x - c is a factor of f(x).
2. If x - c is a factor of f(x), then f(c)0.
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I Synthetic division

• Determine the quotient and the remainder when a
  polynomial function f is divided by g(x) x - c.
• Remainder Theorem
• Examples
• But how to find the quotient. Synthetic division
  finds both simultaneously

Let f be a polynomial function. If f(x) is
divided by x - c, then the remainder is f(c).
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Use synthetic division to find the quotient and
remainder when
x 3 x - (-3)
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f(-3) 278
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Another example
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II Rational Zeros
Rational Zero Theorem Let f be a polynomial
function of degree 1 or higher of the form
where each coefficient is an integer. If p/q in
the lowest terms, is a rational zero of f, then p
must be a factor of a0 and q must be a factor of
an.
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List the potential rational zeros of
p
q
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Find the real zeros of
Factor f over the reals.
There are at most five zeros.
Write factors of -12 and 1 to obtain the
potential rational zeros.
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Thus, -3 is a zero of f and x 3 is a factor of
f.
Thus, -2 is a zero of f and x 2 is a factor of
f.
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Thus f(x) factors as
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• Weve just used synthetic division to find
  rational zeros and factor polynomials.
• More examples

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III Intermediate Value Theorem
Let f denote a continuous function. If altb And if
f(a) and f(b) are of opposite sign, then the
graph of f has at least one zero between a and b.
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Use the Intermediate Value Theorem to show that
the graph of function
has an x-intercept in the interval -3, -2.
f(-3) -11.2 lt 0
f(-2) 1.8 gt 0
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IV Complex numbers and complex zeros of a real
polynomial
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Sum of Complex Numbers
(a bi) (c di) (a c) (b d)i
(2 4i) (-1 6i) (2 - 1) (4 6)i
1 10i
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Difference of Complex Numbers
(a bi) - (c di) (a - c) (b - d)i
(3 i) - (1 - 2i) (3 - 1) (1 - (-2))i
2 3i
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Product of Complex Numbers
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If za bi is a complex number, then its
conjugate, denoted by
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Theorem
The product of a complex number and its conjugate
is a nonnegative real number. Thus if za bi,
then
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Theorem
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If N is a positive real number, we define the
principal square root of -N as
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In the complex number system, the solution of the
quadratic equation
where a, b, and c are real numbers and are given
by the formula
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Solve
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Discriminant of a Quadratic Equation
is called a discriminant
gt0, there are 2 unequal real solutions.

0, there is a repeated real solution.

lt0, there are two complex solutions. The
solutions are conjugates of each other.
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A variable in the complex number system is
referred to as a complex variable.
A complex polynomial function f degree n is a
complex function of the form
A complex number r is called a (complex) zero of
a complex function f if f (r) 0.
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Fundamental Theorem of Algebra
Every complex polynomial function f (x) of degree
n gt 1 has at least one complex zero.
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Theorem
Every complex polynomial function f (x) of degree
n gt 1 can be factored into n linear factors (not
necessarily distinct) of the form
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Conjugate Pairs Theorem
Let f (x) be a complex polynomial whose
coefficients are real numbers. If r a bi is
a zero of f, then the complex conjugate
is also a zero of f.
Corollary
A complex polynomial f of odd degree with real
coefficients has at least one real zero.
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Find a polynomial f of degree 4 whose
coefficients are real numbers and that has zeros
1, 2, and 2i.
f(x)
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Find the complex zeroes of the polynomial function
There are 4 complex zeros.
From Rational Zero Theorem find potential
rational zeros
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Zeros are -2, -1/2, 2 5i, 2 -5i.
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Relevance to previous topics

• Why factor polynomials
• (a) Solve equations of high degree (quadratic
  equation, cubic equation, quadric equation )
• (a) To graph a polynomial function, zeros and
  multiplicity
• (b) To graph a rational function, where its
  zero (need to factor numerator) and where its
  undefined (need to factor denominator)
• (c) To determine whether two polynomials have
  common factors