7.4 and 7.5 Solving and Zeros of Polynomials

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7.4 and 7.5Solving and Zeros of Polynomials

• Day 1

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As with some quadratic equations, factoring a
polynomial equation is one way to find its real
roots.
Recall the Zero Product Property. You can find
the roots, or solutions, of the polynomial
equation P(x) 0 by setting each factor equal to
0 and solving for x.
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Fundamental Theorem of Algebra

• Every polynomial function with degree n greater
  than or equal to 1 has exactly n complex zeros,
  including multiplicities

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The polynomial 3x5 18x4 27x3 0 has two
multiple roots, 0 and 3. The root 0 is a factor
three times because 3x3 0.
The multiplicity of root r is the number of times
that x r is a factor of P(x).
When a real root has even multiplicity, the graph
of y P(x) touches the x-axis but does not cross
it.
When a real root has odd multiplicity greater
than 1, the graph bends as it crosses the
x-axis.
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You cannot always determine the multiplicity of a
root from a graph. It is easiest to determine
multiplicity when the polynomial is in factored
form.
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Example 1A Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
4x6 4x5 24x4 0
4x4(x2 x 6) 0
Factor out the GCF, 4x4.
4x4(x 3)(x 2) 0
Factor the quadratic.
4x4 0 or (x 3) 0 or (x 2) 0
Set each factor equal to 0.
Solve for x.
x 0, x 3, x 2
The roots are 0, 3, and 2.
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Example 1A Continued
Check Use a graph. The roots appear to be located
at x 0, x 3, and x 2. ?
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Example 1B Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x4 25 26x2
x4 26 x2 25 0
Set the equation equal to 0.
Factor the trinomial in quadratic form.
(x2 25)(x2 1) 0
(x 5)(x 5)(x 1)(x 1)
Factor the difference of two squares.
x 5 0, x 5 0, x 1 0, or x 1 0
x 5, x 5, x 1 or x 1
Solve for x.
The roots are 5, 5, 1, and 1.
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Example 2A Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
x3 6x2 12x 8 0
Since this is difficult to factor, use a graph. A
calculator graph shows a bend near (2, 0). ?
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The root 2 has a multiplicity of 3. Therefore x
2 is a factor three times.
Check to see if that is true!
(x 2)(x 2)(x 2) x3 6x2 12x 8
?
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Example 2B Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
x4 8x3 18x2 27 0
A calculator graph shows a bend near (3, 0) and
crosses at (1, 0).
The root 1 has a multiplicity of 1. The root 3
has a multiplicity of 3. Therefore (x 1) is a
factor once, and (x 3) is a factor three times.
Check
(x 1)(x 3)(x 3)(x 3)x4 8x3 18x2 27
?
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Location Principle If P is a polynomial function
and P(x1) and P(x2) have opposite signs, then
there is a real number r between x1 and x2 that
is a zero of P, that is, P(r) 0
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Lesson Quiz
Identify the roots of each equation. State the
multiplicity of each root.
0 and 2 each with multiplicity 2
1. 5x4 20x3 20x2 0
4 with multiplicity 3
2. x3 12x2 48x 64 0
3, 3, 1
3. x3 9 x2 9x
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Homework

• 7.4 ws