Divide using long division Homework Quiz 3.2 Divide using long division ( 𝑥 3 + 3𝑥 2 −𝑥+2)÷(𝑥−1)

3.3 Synthetic Division and Remainder Theorem EQ: How do you use synthetic division and the remainder theorem

Refresher ( 3𝑥 3 +5𝑥+7)÷(𝑥+4)

Synthetic Division Used to simplify long division, when dividing by a linear expression 𝑥−𝑎 Use synthetic division to divide 𝑥 3 −57𝑥+56 by 𝑥−7. What is the quotient and remainder When we have 𝒙−𝟕 we will take −𝟕 and reverse the sign to +𝟕. Then write the coefficients of the polynomial. All the coefficients, including zeros. +7 1 0 −57 56 Step 1: set up synthetic division

+7 1 0 −57 56 1 +7 1 0 −57 56 1 Step 2 bring down first coefficient 1 0 −57 56 1 Step 3 multiply the coefficient by the divisor. Then add to the next coefficient +7 1 0 −57 56 1

Example 1 Divide using Synthetic Division ( 3𝑥 3 +5𝑥+7)÷(𝑥+4)

Example 2 Divide using Synthetic Division ( 2𝑥 3 −5𝑥+7)÷(𝑥−2)

The Remainder Theorem P(x) 𝑥−𝑎 If you divide a polynomial P(x) of degree n≥1 by 𝑥−𝑎, then the remainder is P(a) Example 3 P(x) 𝑥−𝑎 ( 2𝑥 3 −5𝑥+7)÷(𝑥−2) ( 3𝑥 3 +5𝑥+7)÷(𝑥+4)

Example 4 Given the P(x) = 𝑥 5 − 3𝑥 4 − 28𝑥 3 +5𝑥+20, what is P(-4)

Exit Ticket If the polynomial P(x) is divided by x-a and the remainder is 0, what conclusion can we make?

Homework p. 308-309 #23-27 odd, 35-39 odd, 41, 42