Synthetic Division

Show your work for each problem.

1.
When dividing by , what value of do you use in synthetic division?
1. a)
2. b)
3. c)
4. d)
Answer:
For , the divisor is already in the form , so . We use the positive value because we're looking at what makes the divisor zero: means .
2.
When dividing by , what value of do you use?
1. a)
2. b)
3. c)
4. d)
Answer:
, so . Think of it as: what value of makes the divisor zero? means .
3.
What coefficients would you write for in synthetic division? (Write as: a, b, c, d)
Answer: 1, 0, -4, 7
. The coefficients are . We must include for the missing term.
4.
If the original polynomial is degree 4, what degree is the quotient after dividing by ?
1. a)Degree 5
2. b)Degree 4
3. c)Degree 3
4. d)Degree 2
Answer: Degree 3
Dividing a polynomial of degree by a linear factor always gives a quotient of degree . So degree 4 divided by degree 1 gives degree 3.
5.
Use synthetic division: . What is the quotient?
Answer: x + 3
With and coefficients : Quotient: , Remainder:
6.
Divide by using synthetic division.
Answer: x^2 + x - 2
1. What value of do you use for ? 3
2. After bringing down and multiplying by , what do you add to ? 1
3. After multiplying by and adding to , what do you get? -2
4. What is the remainder? 0
7.
Find the remainder when is divided by .
Answer: 17
Using synthetic division with : Remainder = . Also: .
8.
Is a factor of ?
1. a)Yes, remainder is 0
2. b)No, remainder is -6
3. c)No, remainder is 1
4. d)No, remainder is 6
Answer: Yes, remainder is 0
With : Result: . The remainder is , so IS a factor.
9.
Divide by using synthetic division.
Answer: x^3 + x^2 + x + 1
1. What are the coefficients including zeros for missing terms? 1, 0, 0, 0, -1
2. Using , what is (second column)? 1
3. What is (third column)? 1
4. What is the remainder? 0
10.
Divide by . What is the remainder?
Answer: 7
Quotient: , Remainder:
11.
For which divisor can you NOT use synthetic division directly?
1. a)
2. b)
3. c)
4. d)
Answer:
has coefficient for , not . Synthetic division only works directly for where the coefficient of is . For , you'd need long division or factor out the first.
12.
Use synthetic division to evaluate where .
Answer: 23
1. What value of do you use to find ? 4
2. After the first multiplication and addition: 1
3. After the second: 6
4. What is the remainder (which equals )? 23
13.
Find the quotient:
Answer: x^3 - x^2 - x - 11
With : Quotient: , Remainder:
14.
Find all values of such that is a factor of .
Answer: 2
1. For to be a factor, what must the remainder equal? 0
2. Using the Remainder Theorem, what must equal? 0
3. Substitute : . Simplify to find . 2
15.
If the synthetic division of by gives remainder , and division by gives remainder , what can you conclude?
1. a) has no roots
2. b) equals
3. c)
4. d) is a factor of
Answer: is a factor of
If is a factor and is a factor, then their product is also a factor of . This means has roots at and .
16.
Factor completely by first using synthetic division to find one root.
Answer: (x - 1)(x - 2)(x - 3)
1. Test : What is ? 0
2. Since , what linear factor do you know? (x - 1)
3. After dividing by , what quadratic do you get? x^2 - 5x + 6
4. Factor . (x - 2)(x - 3)
17.
If has remainder and quotient , find .
Answer: -9
. So and , giving . Wait, let me recalculate: we need to match , so and . Therefore .
18.
Use synthetic division twice to divide by and then by .
Answer: x^2 - 2x - 3
1. First divide by . What is the quotient? x^3 - 4x^2 + x + 6
2. What is the remainder after dividing by ? 0
3. Now divide by . What is the quotient? x^2 - 2x - 3
4. What is the remainder after this second division? 0