Lesson 3.4

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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 no
common factors, is a root of the equation, then
p is a factor of the constant term and q is a
factor of the leading coefficient.
Ex. 1
List all possible roots of
Then determine the rational roots.
List possible values of p
List possible values of q
Possible rational roots
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You Try

• List all possible rational zeros of
• f(x) x3 2x2 5x 6
• Possible values of p
• Possible values of q
• Possible rational roots(p/q)

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Finding Zeros of a Polynomial Function

• Now, use synthetic division to test and find the
  roots/factors. The last number must be a zero to
  show the root is a factor. Degree is 3, so there
  should be 3 solutions.
• Possible rational roots

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Checking with Synthetic Division

• 1 6 11 -3 -2 1 is not a zero
• 6 17 14 because the
• 6 17 14 12 remainder does
• not equal 0!!
• Now lets try -2.
• -2 6 11 -3 -2
• -12 2 2 -2 is a
  zero!!!
• 6 -1 -1 0

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Finding the Zero (cont.)

• Take -2 and write it as a factor which is x 2
  and take your answer from synthetic division and
  put it into a polynomial 6x2 x -1.
• Now factor 6x2 x -1
• (2x 1 )(3x 1)
• Now put all the factors together
• (x2)(2x-1)(3x1).
• Put factors equal to zero to find the zeros.
• X -2, ½, -1/3 (3 real rational solutions)

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• The process

Dont forget Step 1Find your ps and qs and
list all possible roots. Step 2Number of
roots/zeros is based on highest degree. Use
synthetic division to find your first root. If
that does not work, USE YOUR CALCULATOR!!!
Remember your multiplicity ideas as well. If
the polynomial crosses the x axis, the
multiplicity is odd. If the polynomial touches
and turns around, it is even. Step3 After
finding a root, factor the rest on your own. If
not factorable, use the quadratic formula. Step
4 Then, solve for the rest of the roots. Roots
can be real or imaginary. If the roots are
imaginary, then they occur in conjugate pairs!
To set up factors (in parenthesis) just change
their signs.
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You Try!!

• Find all zeros of f(x) x3 7x2 11x 3
• Step 1 Find possible rational roots.
• p q possible
  rational roots
• Use synthetic division to find one rational root
  or by the calculator. By using the calculator,
  find one zero. Show on the calculator to class.
• Hint You will need to use the quadratic
  formula?
• One root is 3 from calculator. Now find the
  other roots.
• How many should there be?

3
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Answer

• The solution set is -3,-2 - v5, -2v5
• Your solutions can be imaginary or real. If your
  solution is imaginary, it will be written as a
  complex conjugate. If it is real, it could be
  rational (nice numbers) or irrational (not nice
  numbers).

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You Try Again

• Solve x4 6x3 22x2 30x 13
• Use Calculator to find two zeros.
• Answer 1,2-3i,23i

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Zeros of Polynomial Functions
Complex Numbers (abi)
Imaginary Numbers (bi)
REAL number system
Rational Numbers
Irrational Numbers
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General shapes of graphs with a positive leading
coefficient.
Degree 2
Degree 3

• Degree 1

1 zero
2 zeros
3 zeros
Degree 5
Degree 4
5 zeros
4 zeros
Remember, zeros are just x-intercepts.
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Finding a Polynomial Function with Given Zeros
EXAMPLE 1 Find a 3rd degree polynomial function
f(x) with real coefficients that has -3 and i as
zeros and such that f(1) 8.
f(x) an (x-c1)(x-c2)(x-c3)
Now substitute in the zeros with what you know.
Do not forget about the conjugate pairs.
f(x) an(x3)(xi)(x-i)
Multiply the polynomial out.
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f(x) an(x3 3x2 x 3)
f(1) an(1)3 3(1)2 1 3 8an
8 8an
f(1) 8
an 1
Polynomial Equation is f(x) (x3) (x21) or x3
3x2 x 3

• YOU TRY Find a 3rd degree polynomial function
  f(x) with real coefficients that has 4 and 2i as
  zeros and such that f(-1) -50.

Answer f(x) 2x3 8x2 8x -32
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Summary

• Describe how to find the possible rational zeros
  of a polynomial function.