Precalculus PreAP/Dual, Revised ©2017 Rational Zero Test Section 2.5 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/3/2018 10:34 AM §2.5: Rational Zero Test

Finding Rational Zeros Where the graphs cross the 𝒙–axis (𝒚 is zero) Known as zeros or roots To find the real–number zeros, it can be rational, irrational and/or imaginary numbers To find out how many zeros the graph has, the easiest way to find the amount of degrees the graphs has 11/3/2018 10:34 AM §2.5: Rational Zero Test

Steps Identify the constant term and coefficient of the polynomial List all multiples of each number and don’t forget the plus-or-minus symbol for each polynomial Apply the equation: 𝒑 𝒒 = Factors of Constant Term Factors of Leading Coefficient List ALL of the possible rational zeros 11/3/2018 10:34 AM §2.5: Rational Zero Test

What multiplies with what equals that number? Example 1 List all possible rational zeros for 𝒙 𝟖 +𝟏𝟖 𝒙 𝟒 − 𝟗𝒙 𝟐 +𝟐=𝟎 What multiplies with what equals that number? 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 2 List all possible rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 11/3/2018 10:34 AM §2.5: Rational Zero Test

Your Turn List all possible rational zeros for 𝟒 𝒙 𝟒 −𝟐𝟏 𝒙 𝟑 +𝟏𝟖 𝒙 𝟐 +𝟏𝟗𝒙−𝟔=𝟎 11/3/2018 10:34 AM §2.5: Rational Zero Test

Steps Determine what the leading coefficient and constant term are. Put it in the rational zero test 𝒑 𝒒 Use synthetic division to find which factor gives the remainder of zero. If the polynomial cannot be factored, repeat steps 2 and 3 again Factor or use the Quadratic Formula to get the leftover polynomial out. Equal it all to zero and put it in 𝒙= form. 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 3 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟐𝒙−𝟐 Step 1: What are the factors of the constant term and the leading coefficient? Constant Term (the number without the letter): Coefficient: (the number in front of the letter with the highest degree) Step 2: Put it in rational–zero test 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 3 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟐𝒙−𝟐 Step 3: Test each possible rational form 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 3 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟐𝒙−𝟐 Step 4: Factor or use the Quadratic Formula to get the leftover polynomial out. 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 3 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟐𝒙−𝟐 Step 5: Equal it all to zero and put it in 𝒙= form. 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 3 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟐𝒙−𝟐 Step 6: Use graphing calculator to check 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 4 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟑 𝒙 𝟑 +𝟕 𝒙 𝟐 −𝟐𝟐𝒙−𝟖 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 5 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟑 −𝟕 𝒙 𝟐 +𝟏𝟏𝒙−𝟓 11/3/2018 10:34 AM §2.5: Rational Zero Test

Your Turn Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟔 𝒙 𝟑 + 𝒙 𝟐 −𝟏𝟎𝒙+𝟑 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 1: What are the factors of the constant term and the leading coefficient? Step 2: Apply p/q 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 3: Test each possible rational form 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 3: Test each possible rational form 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 4: Identify the equation & Solve 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 4: Identify the equation & Solve 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 6 Find the real and imaginary rational zeros for 𝒇 𝒙 =𝟐 𝒙 𝟒 + 𝒙 𝟑 −𝟏𝟕 𝒙 𝟐 −𝟒𝒙+𝟔 Step 5: Check 11/3/2018 10:34 AM §2.5: Rational Zero Test

Example 7 Find the real and imaginary rational zeros for 𝒇 𝒙 = 𝒙 𝟒 − 𝒙 𝟑 − 𝒙 𝟐 −𝒙−𝟐 11/3/2018 10:34 AM §2.5: Rational Zero Test

Assignment Worksheet 11/3/2018 10:34 AM §2.5: Rational Zero Test