MM3A3: Students will solve a variety of equations and inequalities. Essential Questions How do we apply various theorems to find roots of polynomial functions and to solve polynomial equations? Standards MM3A3: Students will solve a variety of equations and inequalities. MM3A3a: Find real and complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem, and fundamental theorem of algebra, incorporating complex and radical conjugates. MM3A3b: Solve polynomial, exponential, and logarithmic equations analytically, graphically, and using appropriate technology. MM3A3c: Solve polynomial, exponential, and logarithmic inequalities analytically, graphically, and using appropriate technology. Represent solution sets of inequalities using interval notation. MM3A3d: Solve a variety of types of equations by appropriate means, choosing among mental calculation, pencil and paper, or appropriate technology.  

Remember the steps to long division problems? Check this problem out. Any questions about your long division problems from yesterday?

Use synthetic division to divide (2x3 – 36x + 8) by (x – 4). 4 2 0 -36 8

Use synthetic division to divide (2x3 – 36x + 8) by (x – 4). 4 2 0 -36 8 8 32 -16 2 8 -4 -8 2x2 + 8x – 4 + You get the same answer.

Let’s practice synthetic division before we tackle how to solve cubic polynomials in general.   4. Do the following division problems synthetically. a. b. c.

Let’s practice synthetic division before we tackle how to solve cubic polynomials in general.   4. Do the following division problems synthetically. a. b. c. 2 10 -17 -7 2 -4 1 3 -10 -24 -1 1 0 -7 -8 20 6 -2 -4 4 24 -1 1 6 10 3 -1 0 1 -1 -6 0 1 -1 -6 -2 10x2 + 3x – 1 x2 – x – 6 x2 – x – 6

5. Use synthetic division to find the quotient and remainder for each division described below. Use the Remainder Theorem to determine the value of f(t) for each polynomial and divisor. Use the Factor Theorem to determine whether (x – t) is a factor of the polynomial.   a. f(x) = x3 – 4x2 – 29x – 24 b. g(x) = x5 – 5x4 – 2x3 + 10x2 + x – 10 divisor: (x + 3) divisor: (x – 5) c. h(x) = 2x3 – 7x2 + 4x + 4 d. j(x) = x6 + x5 + x4 – x3 – 2x + 22 divisor: (x – ½) divisor: (x – 1)

1) f(x) = -x3 + 6x – 7 at x = 2 2) f(x) = x3 + x2 – 5x -6 at x = 2 The Remainder Theorem Evaluate each function at the given value. 1) f(x) = -x3 + 6x – 7 at x = 2 2) f(x) = x3 + x2 – 5x -6 at x = 2 3) f(a) = a3 + 3a2 + 2a + 8 at a = -3 4) f(a) = a3 + 5a2 + 10a + 12 at a = -2

5) f(a) = a4 + 3a3 – 17a2 + 2a -7 at a = 3 6) f(x) = x5 – 47x3 – 16 x2 + 8x + 52 at x = 7 State if the given binomial is a factor of the given polynomial. 7) (k3 – k2 – k – 2) / (k – 2) 8) f(x) = (b4 – 8b3 – b2 + 62b – 34) / (b – 7)

(n4 + 9n3 + 14n2 + 50n + 9) / (n + 8) 10) (p4 + 6p3 + 11p2 + 29p – 13) / (p + 5) 11) (p4 – 8p3 + 10p2 + 2p + 4) / (p – 2) 12) (n5 – 25n3 – 7n2 – 37n – 18) / (n + 5)

(x5 + 6x4 – 3x2 – 22x – 29) / (x + 6) 14) (n4 + 10n3 + 21n2 + 6n – 8) / (n + 2) 15) (-8x4 + 36x3 + 14x2 + 25x + 25) / (x – 5) 16) (x4 + 2x3 – 8x2 – 11x + 13) / (x +3)

(r3 + 2r2 – 33r + 7) / (r + 7) 18) (r4 – 5r3 – 20r2 – 4r + 10) / (r + 2) 19) (p5 + 8p4 + 2p2 + 19p + 16) / (p + 8) 20) (x4 – 2x3 – 16x2 + 28x +9) / (x – 4)

(r5 + 6r4 – 13r3 – 5r2 – 8r + 14) / (r – 2) 22) (8v5 + 32v4 + 5v + 20) / (v + 4)