6.5 finding real roots of polynomial equations. -factoring to find real roots -explain multiplicity...

6.5 Finding Real Roots of Polynomial Equations

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6.5

Finding Real Roots of Polynomial Equations

-factoring to find real roots-explain Multiplicity

The root 0 has a multiplicity of 3.The root -3 has a multiplicity of 2.

5 4 33 18 27 0x x x

Page 3: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

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

Page 4: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

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.

Page 5: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Rational Root Theorem

• Use the Rational Root Theorem to identify all possible rational roots.

3 23 2 1 0x x

Page 6: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Rational Root Theorem

• Use the Rational Root Theorem to identify all possible rational roots.

32 10 0x x

Page 7: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

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!!!

Page 8: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Find all real roots.3 23 4 17 6 0x x x

Page 9: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Find all real roots.3 22 3 5 2 0x x x

Page 10: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Find all real roots.4 3 24 7 34 24 0x x x x

Page 11: 6.5 Finding Real Roots of Polynomial Equations. -factoring to find real roots -explain Multiplicity A. The root 0 has a multiplicity of 3. The root -3

Homework

• Page 442 (2-9, 11-14)

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