Section 5.4 Day 2: Dividing Polynomials
Name: ____________________
Learning Goal: I will be able to use long division to divide polynomials with remainders.
I will be able to divide polynomials using synthetic division.
PROBLEM TYPE 1: Dividing Polynomials
Complete the given division problem using synthetic division.
1) (𝑥 3 − 14𝑥 2 + 51𝑥 − 54) ÷ (𝑥 + 2)
Step 1: Reverse the sign of +2. Write the coefficients of the polynomial in descending order.
-2
1
-14
51
-54
Step 2: Bring down the first coefficient.
-2
1
-14
51
-54
1
Step 3: Multiply the coefficient by the divisor. Add to the next coefficient.
-2
1
1
-14
-2
-16
51
-54
Step 4: Continue multiplying and adding through the last coefficient.
-2
1
-14
51
-54
-2
32
-166
1
-16
83
-220
2
The quotient is 𝑥 − 16𝑥 + 83, 𝑅 − 220
Complete the given division problems using synthetic division.
1. (𝑥 3 − 8𝑥 2 + 17𝑥 − 10) ÷ (𝑥 − 5)
2. (𝑥 3 − 57𝑥 + 56) ÷ (𝑥 − 7)
3. (2𝑥 4 + 23𝑥 3 + 60𝑥 2 − 125𝑥 − 500) ÷ (𝑥 + 4)
PROBLEM TYPE 2: Evaluating a Polynomial
REMAINDER THEOREM: This theorem provides a quick way to find the remainder of a polynomial long-division
problem.
***If you divide a polynomial P(x) of a degree n≥1 by x – a, then the remainder is P(a).***
5) Given that P(x) = 𝑥 5 − 2𝑥 3 − 𝑥 2 + 2, what is the remainder when P(x) is divided by x - 3.
Synthetic Division
Evaluate P(3)
6) Given that P(x) = 𝑥 45 + 20, what is the remainder when P(x) is divided by x + 1.
Problem Type 3: Factoring a Polynomial
Use synthetic division and the given factor to completely factor each polynomial function.
7) 𝑦 = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6; (𝑥 + 1)
8) 𝑦 = 𝑥 3 + 6𝑥 2 − 𝑥 − 30; (𝑥 + 3)
Problem Type 4: Is this binomial a factor?!?
Use synthetic division to determine if the given binomial is a factor of the polynomial. If you complete synthetic
division with the given binomial and the remainder is 0, then the given binomial is a factor!
9) 𝑦 = 𝑥 3 + 8𝑥 2 + 11𝑥 − 20; (𝑥 − 1)
10) 𝑦 = 𝑥 3 + 6𝑥 2 − 9𝑥 − 14; (𝑥 + 2)