CC Algebra 2
Name: ________________
Module 1 Lesson 18: Overcoming a Second Obstacle in Factoring—
What If There is a Remainder?
Opening Exercise
Write the rational number
13
4
as a mixed number.
Method 1:
Method 2:
Method 3:
Example 1
a. Find the quotient by factoring the numerator.
b. Find the quotient.
𝑥 2 +3𝑥+2
𝑥+2
𝑥 2 +3𝑥+5
𝑥+2
Method 1:
How could we use the answer from a. to evaluate this quotient?
Method 2:
Reverse Tabular
Method 3:
Long Division
Method 4:
Synthetic Division
Description—a shorthand method of dividing polynomials; divisor has to be a __________ binomial with
coefficient of _______. In order for synthetic division to work, the polynomial must be written in
_______________ form, using ____ as a coefficient for any missing terms, and the divisor must be in the
form _____________.
Divide using synthetic division:
The Process:
1. Write the coefficients of the dividend in a row.
Write the value of a for the divisor in front of the
row of coefficients in a “box”. Draw a “T”
Bring the first coefficient down (below the “T”).
2. Multiply the first coefficient by the divisor and
write the product under the next coefficient.
Combine the coefficient and the product (watch
the signs).
3. Repeat step 2 until all combining has been
completed.
4. Each coefficient of the row under the “T” represents
the quotient one degree less than the dividend. The
“box” formed by the “T” represents the remainder.
𝑥 2 +3𝑥+5
𝑥+2
Example 2
a. Find the quotient by factoring the numerator.
b. Find the quotient.
𝑥 3 −8
𝑥−2
𝑥 3 −4
𝑥−2
Exercises 1-10
Find each quotient by inspection.
1.
𝑥+4
𝑥+1
2.
2𝑥−7
𝑥−3
3.
𝑥 2 −21
𝑥+4
Find each quotient by using the reverse tabular method or synthetic division.
4.
𝑥 2 +4𝑥+10
𝑥−8
5.
𝑥 3 −𝑥 2 +3𝑥−1
𝑥+3
6.
𝑥 2 −2𝑥−19
𝑥−1
Find each quotient by using long division.
7.
𝑥 2 −𝑥−25
𝑥+6
8.
𝑥 4 −8𝑥 2 +12
𝑥+2
9.
4𝑥 3 +5𝑥−8
2𝑥−5
Rewrite the numerator in the form (𝑥 − ℎ)2 + 𝑘 by completing the square. Then find
the quotient.
10.
𝑥 2 +4𝑥−9
𝑥+2