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6.5
Finding Real Roots of Polynomial Equations
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-factoring to find real roots-explain Multiplicity
A.
The root 0 has a multiplicity of 3.The root -3 has a multiplicity of 2.
5 4 33 18 27 0x x x
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Multiple Roots
• Multiple roots appear in a polynomial equation that has a factor that appears more than once.
Graph on your calculator.
Notice where it touches the x-axis and what it looks like.
5 4 33 18 27 0x x x
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Rational Root Theorem
• Every rational root can be written in the form
• Where p is a factor of the constant term
• q is a factor of the leading coefficient.
p
q
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Rational Root Theorem
• Use the Rational Root Theorem to identify all possible rational roots.
3 23 2 1 0x x
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Rational Root Theorem
• Use the Rational Root Theorem to identify all possible rational roots.
32 10 0x x
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Find all the real roots of each equation.
• Steps:1. List all possible roots
2. Use synthetic division until you get to a quadratic equation
3. Write the factors ( ) ( )
4. Write the zeros
These steps MUST be shown on all problems!!!
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Find all real roots.3 23 4 17 6 0x x x
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Find all real roots.3 22 3 5 2 0x x x
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Find all real roots.4 3 24 7 34 24 0x x x x
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Homework
• Page 442 (2-9, 11-14)