2.6 Solving Polynomial Equations

1. 2.6 Solving Polynomial Equations

2. Now that we are pros at solving quadratics, we will raise the bar & solve any polynomial equation - Remember answers can be real & non-real - We will utilize our factoring skills & use the quadratic formula • Note: the largest power of x indicates how many answers/ zeros/ roots the equation will have - Each equation is a tad bit different: just use your skills!  Ex 1) Find the zeros: f (x) = –3x4 + 18x3 – 27x2 0 = –3x4 + 18x3 – 27x2 0 = –3x2(x2 – 6x + 9) 0 = –3x2(x – 3)2 x = 0, 3 (but you need 4 answers!) so… 0(mult 2), 3(mult 2)

3. 0 = 4x7 – 49x5 0 = x5(4x2 – 49) 0 = x5(2x – 7)(2x + 7) Ex 2) Find the zeros: f (x) = 4x7 – 49x5 x = 0 (mult 5), Ex 3) Solve: g(x) = 3x4 – 147 0 = 3x4 – 147 0 = 3(x4 – 49) 0 = 3(x2 – 7)(x2 + 7) x2 – 7 = 0 x2 + 7 = 0 x2 = 7 x2 = –7

4. Song!  Reminder: Sum/ Diff of Cubes a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Ex 4) Solve: f (x) = x6 + 7x3 – 8 0 = x6 + 7x3 – 8 0 = (x3 – 1)(x3 + 8) 0 = (x – 1)(x2 + x + 1)(x + 2)(x2 – 2x + 4) (quad form!) (quad form!) 1 –2

5. 9x2 (3x– 1) – 1(3x– 1) = 0 (9x2 – 1)(3x – 1) = 0 (3x + 1)(3x – 1)(3x – 1) = 0 Ex 5) Solve: 27x3 – 9x2 – 3x +1 = 0 Ex 6) Use a graphing calculator to approximate the real zeros of the function to the nearest hundredth. P(x) = 4x4 + 3x3 – 2x – 1 Just like before: 2nd CALC 2:zero –0.57, 0.75

6. Homework #207 Pg 97 # 1–29 odd, 30, 31, 33, 35, 38