Algebra 2 Practice Test: Polynomial Long and Synthetic Division
- Polynomial Long and Synthetic Division
- 10 questions
- 15 minutes
These 10 questions practice dividing polynomials by long division and synthetic division, reading quotients and remainders, and using the remainder theorem. Before you divide, write a 0 for every missing power of x.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the quotient when x3 + 2x2 − 5x − 6 is divided by x + 1?
Correct answer: B
Synthetic division with −1 gives the row 1, 1, −6 and remainder 0, so the quotient is x2 + x − 6. Using +1 instead of −1 in synthetic division gives a different, wrong row.
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Question 2
You divide 3x4 − 2x2 + 5 by x − 1 using synthetic division. Which list of coefficients should you write in the first row?
Correct answer: D
Every power from x4 down to the constant needs a place, so the missing x3 and x terms get zeros: 3, 0, −2, 0, 5.
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Question 3
What is the remainder when x4 − 3x3 + 2x − 7 is divided by x − 3?
Correct answer: C
By the remainder theorem, the remainder is p(3) = 81 − 81 + 6 − 7 = −1. Evaluating at −3 instead gives p(−3) = 149.
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Question 4
What is the quotient when 6x3 + x2 − 4x + 1 is divided by 2x − 1?
Correct answer: A
Long division: 6x3 ÷ 2x = 3x2, then 2x, then −1, with remainder 0. Synthetic division with 1/2 gives 6x2 + 4x − 2, which must still be divided by 2 because the divisor is 2x − 1, not x − 1/2.
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Question 5
When x3 + 4x2 + 3x + 5 is divided by x2 + 1, what are the quotient and the remainder?
Correct answer: B
(x2 + 1)(x + 4) = x3 + 4x2 + x + 4. Subtracting from the dividend leaves 2x + 1, whose degree is less than 2, so the division stops there.
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Question 6
When the polynomial p(x) is divided by x − 4, the quotient is x2 + 2x − 1 and the remainder is 3. What is p(x)?
Correct answer: D
p(x) = (x − 4)(x2 + 2x − 1) + 3 = x3 + 2x2 − x − 4x2 − 8x + 4 + 3 = x3 − 2x2 − 9x + 7. Forgetting to add the remainder 3 changes the constant term.
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Question 7
Synthetic division of a cubic polynomial by x + 3 ends with this bottom row:
2 −1 4 | −17
What are the quotient and the remainder?
Correct answer: A
Dividing a cubic by a linear factor gives a quadratic, so 2, −1, 4 are the coefficients of 2x2 − x + 4. The last number in the row, −17, is the remainder.
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Question 8
What is the quotient when x3 + 8 is divided by x + 2?
Correct answer: A
With coefficients 1, 0, 0, 8 and the number −2, synthetic division gives 1, −2, 4 and remainder 0. This matches the sum of cubes pattern a3 + b3 = (a + b)(a2 − ab + b2).
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Question 9
When x3 + kx2 − 4x + 1 is divided by x − 2, the remainder is 9. What is the value of k?
Correct answer: C
The remainder equals p(2) = 8 + 4k − 8 + 1 = 4k + 1. Setting 4k + 1 = 9 gives k = 2.
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Question 10
A box has a volume of x3 + 7x2 + 14x + 8 cubic units and a height of x + 1 units. Which expression gives the area of its base?
Correct answer: B
Base area = volume ÷ height. Synthetic division with −1 gives 1, 6, 8 and remainder 0, so the base area is x2 + 6x + 8 = (x + 2)(x + 4).
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