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6.5 Theorems About Roots of Polynomial Equations

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## 6.5 Theorems About Roots of Polynomial Equations

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**6.5 Theorems About Roots of Polynomial Equations**6.5.1 Rational Root Theorem**6.5.1: Rational Root Theorem**• To find rational roots of an equation, you must divide the factors of the constant, by the factors of the leading coefficient • Factors of the constant (p) • Factors of the leading coefficient (q) • Possibilities are:**Example 1: Finding Rational Roots**Find the rational roots of 3x3 – x2 – 15x + 5 = 0 Step 1: List the possible rational roots. So the possibilities are:**Example 1 Continued**3x3 – x2 – 15x + 5 = 0 Step 2: Test each possible rational root. 3( )3 – ( )2 – 15( ) + 5 = 0 1 -8 1 1 ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ 3( )3 – ( )2 – 15( ) + 5 = 0 -1 16 -1 -1 0 3( )3 – ( )2 – 15( ) + 5 = 0 1/3 1/3 1/3 88/9 3( )3 – ( )2 – 15( ) + 5 = 0 -1/3 -1/3 -1/3 280 3( )3 – ( )2 – 15( ) + 5 = 0 5 5 5 3( )3 – ( )2 – 15( ) + 5 = 0 -320 -5 -5 -5 -80/9 3( )3 – ( )2 – 15( ) + 5 = 0 5/3 5/3 5/3 3( )3 – ( )2 – 15( ) + 5 = 0 30 -5/3 -5/3 -5/3**Find the roots of 2x3 – x2 + 2x – 1 = 0**Step 1: List the possible rational roots. Step 2: Test each possible rational root until you find a root Step 3: Use synthetic division with the root you found in Step 2 Step 4: Find the rest of the roots by solving (use quadratic formula if necessary) +1, +1/2 2(1)3 –(1)2 + 2(1) – 1= 2(-1)3 –(-1)2 + 2(-1) – 1= 2(1/2)3 –(1/2)2 + 2(1/2) – 1= Example 2: Using the Rational Root Theorem**6.5 Theorems About Roots of Polynomial Equations**6.5.2 Irrational Root & Imaginary Root Theorem**Irrational Root Theorem**• If is a root, then it’s conjugate is also a root**Example3: Finding Irrational Roots**• A polynomial equation with rational coefficients has the roots ___________ and ___________. Find two additional roots. The additional roots are: ____________ and _______________**Imaginary Root Theorem**• If is a root, then it’s conjugate is also a root**Example 4: Finding Imaginary Roots**• A polynomial equation with real coefficients has the roots 2 + 9i and 7i. Find two additional roots. • The additional roots are:______ and _____**Find a third-degree polynomial equation with rational**coefficients that has roots of 3 and 1 + i. Step 1: Find the other imaginary root Step 2: Write the roots in factored form Step 3: Multiply the factors Example 5: Writing a Polynomial Equation from Its Roots