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Quizzes > High School Quizzes > Mathematics

Worksheet Practice Quiz: Long Division of Polynomials

Ace polynomial division with focused practice questions

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Polynomial Division Showdown quiz paper art engaging high school algebra students in trivia.

In polynomial long division, which term do you divide first?
The last term of the dividend by the first term of the divisor
The entire dividend by the divisor as a whole
The constant term of the dividend by the constant term of the divisor
The leading term of the dividend by the leading term of the divisor
The first step in polynomial long division is to divide the leading term of the dividend by the leading term of the divisor. This determines the first term of the quotient.
When performing polynomial long division, what does the remainder represent?
The polynomial left after subtracting the product of the divisor and quotient from the dividend
The result of multiplying the divisor by the quotient
The factor that completes the dividend
The constant term of the dividend
The remainder is the part of the dividend that is left over after subtracting the product of the divisor and the quotient. It represents the portion that cannot be evenly divided by the divisor.
Which operation is performed immediately after determining the quotient term in polynomial long division?
Addition of the quotient term to the dividend
Division of the quotient term by the divisor
Multiplication of the divisor by the quotient term
Subtraction of the divisor from the dividend
After determining the quotient term, you multiply the divisor by this term and then subtract the result from the dividend. This step is crucial for reducing the dividend for subsequent division steps.
In polynomial long division, what is the next step after subtracting the product of the divisor and quotient term?
Divide the divisor by the remainder
Combine like terms in the quotient
Multiply the remainder by the divisor
Bring down the next term of the dividend to continue the division
After the subtraction, you bring down the next term of the dividend to continue the division process. This step allows the division procedure to proceed methodically through all terms.
How is the polynomial long division algorithm best described?
A step-by-step procedure that divides the dividend by the divisor by aligning like terms
A method for factoring polynomials without performing division
A technique used primarily for solving quadratic equations
A process that involves multiplying polynomials together
Polynomial long division is a systematic method that divides a dividend by a divisor while aligning like terms to ensure accurate subtraction at each step. This algorithm helps simplify polynomials and find quotients and remainders.
Divide 2x³ + 3x² - 5x + 6 by (x - 1) using long division. What is the quotient?
2x² + 5x
2x² - 3x + 6
2x² - x - 6
2x² + 5x + 6
Dividing 2x³ by x gives 2x² and following the long division steps leads to a quotient of 2x² + 5x, with a remainder of 6. The quotient is considered separately from the remainder.
Find the remainder when dividing x³ - 3x² + 4 by (x - 2).
-2
0
4
2
Using the Remainder Theorem, substitute x = 2 into the polynomial: 2³ - 3(2²) + 4 = 8 - 12 + 4 = 0. Thus, the remainder is 0.
Divide 3x³ + 5x² - 2x + 1 by (x + 2) using long division. What is the remainder?
-1
0
2
1
By applying the Remainder Theorem with x = -2, substitute into the polynomial to get: 3(-2)³ + 5(-2)² - 2(-2) + 1, which simplifies to 1. Therefore, the remainder is 1.
Which theorem allows you to determine the remainder when dividing a polynomial by (x - c) without carrying out the full division?
The Remainder Theorem
The Fundamental Theorem of Algebra
The Division Algorithm
The Factor Theorem
The Remainder Theorem states that the remainder of dividing a polynomial by (x - c) is equal to the value of the polynomial at x = c. This provides a quick method to find the remainder.
Why is it important to align like terms when performing polynomial long division?
To easily determine the leading coefficient
To simplify the polynomial into its prime factors
To ensure the correct subtraction of corresponding terms
To factor the polynomial completely
Aligning like terms ensures that terms of the same degree are subtracted correctly. This alignment minimizes errors and maintains the proper structure required for accurate division.
When dividing x³ by (x - 3), what is the first term of the quotient?
x
x² + 3
x²
x² - 3
Dividing the leading term x³ by x (from x - 3) yields x², which serves as the first term in the quotient. This step is the starting point in the long division process.
Perform the long division of 5x² + 7x + 3 by (x + 1). What is the quotient?
5x + 1
x + 2
5x + 3
5x + 2
Dividing 5x² by x gives 5x, and further steps in the long division process yield an additional term of 2, resulting in a quotient of 5x + 2. The remainder is handled separately.
What should be done if a term in the dividend is missing when setting up polynomial long division?
Insert a term with a zero coefficient for that missing degree
Rearrange the polynomial terms
Skip the missing term and proceed
Combine the adjacent terms
Inserting a term with a zero coefficient maintains the proper alignment of like terms. This technique ensures accuracy during the subtraction step of the long division process.
Which expression correctly represents the Division Algorithm for polynomials?
Dividend = Divisor - Remainder
Dividend = (Divisor - Quotient) + Remainder
Dividend = Divisor + Quotient
Dividend = Quotient - Remainder + Divisor
The Division Algorithm states that any dividend can be written as the product of the divisor and the quotient, plus the remainder. This is a fundamental principle in polynomial division.
If the degree of the dividend is less than the degree of the divisor, what is the result of the division?
Quotient = 0 and Remainder = dividend
Quotient = dividend and Remainder = 0
Quotient = dividend and Remainder = dividend
Both Quotient and Remainder are 0
When the dividend's degree is lower than that of the divisor, division cannot proceed further, resulting in a quotient of 0. The entire dividend is then considered as the remainder.
Divide 6x❴ - 5x³ + 4x² - 3x + 2 by (2x² - x + 1) using long division. What is the quotient?
3x² - x
3x² + x
3x² - 2x
3x² - x + 1
Dividing the leading term 6x❴ by 2x² yields 3x². Continuing the long division process produces -x as the next term, resulting in a quotient of 3x² - x. The remainder is handled separately.
Divide 4x❴ + 0x³ - x² + 2x - 5 by (2x² + x - 3) using long division. What is the remainder?
4x - 4
4x + 4
-4x + 4
-4x - 4
Following the long division procedure by dividing the highest degree terms repeatedly produces a remainder of -4x + 4. This result represents the portion of the dividend that cannot be divided by the divisor.
In polynomial division, if P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder, which equation correctly represents their relationship?
P(x) = Q(x) + D(x) · R(x)
P(x) = D(x) + Q(x) - R(x)
P(x) = D(x) · (Q(x) + R(x))
P(x) = D(x) · Q(x) + R(x)
The Division Algorithm for polynomials states that the dividend P(x) equals the divisor D(x) multiplied by the quotient Q(x), plus the remainder R(x). This relationship is foundational in polynomial division.
When a dividend polynomial is missing a term for a specific degree, what is the recommended practice during long division?
Merge adjacent terms to fill the gap
Skip the missing term
Insert a term with a zero coefficient as a placeholder
Rearrange the polynomial in descending order
Inserting a placeholder with a zero coefficient ensures that like terms are properly aligned during subtraction. This practice prevents mistakes and maintains the structure of the division process.
If you divide a polynomial of degree 5 by a polynomial of degree 3, what can be said about the degrees of the quotient and the remainder?
Both the quotient and the remainder will have degree 3
The quotient will have degree 3 and the remainder will have degree 2
The quotient will have degree 5 and the remainder will be zero
The quotient will have degree 2 and the remainder will have a degree less than 3
The degree of the quotient is the degree of the dividend minus the degree of the divisor, which in this case is 5 - 3 = 2. Moreover, the degree of the remainder must be less than the degree of the divisor, so it is less than 3.
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Study Outcomes

  1. Understand the fundamental principles of polynomial long division.
  2. Apply the long division algorithm to simplify polynomial expressions.
  3. Analyze the division steps to identify potential calculation errors.
  4. Calculate and interpret the remainder in polynomial division problems.
  5. Synthesize the overall division process to verify and validate results.

Worksheet: Long Division of Polynomials Cheat Sheet

  1. Master Long Division - Long division of polynomials is like detective work: you divide, multiply, subtract, and repeat until the remainder is lower degree than the divisor. It might feel tedious at first, but aligning terms properly turns it into a smooth, almost meditative groove. Use this method to tame even the trickiest polynomial beasts. Polynomials - Long Division
  2. Embrace Synthetic Division - Synthetic division is your fast‑track ticket when the divisor is a nice, neat linear term of the form (x − k). With just a handful of additions and multiplications, you'll breeze through calculations in record time. It's perfect for those "quick answer" moments on homework or exams. Synthetic Division - Wolfram MathWorld
  3. Apply the Remainder Theorem - If you divide a polynomial P(x) by (x − k), the remainder is simply P(k). No need for a full division algorithm - just plug in and get your answer in two seconds flat. This little trick is a huge time‑saver for checking remainders on the fly. Synthetic Division of Polynomials with Examples
  4. Utilize the Factor Theorem - The Factor Theorem and the Remainder Theorem are best friends: if P(k) = 0 then (x − k) is a factor of P(x). This insight turns root‑finding into a mini‑celebration: test a candidate, and if it works, you've factored one piece. It's the shortcut every polynomial‑solver dreams about. Synthetic Division of Polynomials with Examples
  5. Divide by Monomials First - When dividing by a monomial, simply split your polynomial term by term. This breakdown makes the process feel like slicing cake: one piece at a time until it's all gone. It's a foundational skill that builds confidence for more complex divisions. Dividing Polynomials - Long Division Method
  6. Factor for Binomial Divisions - If you can factor your polynomial to reveal a common binomial, cancellation is your best friend. Factorization before division can turn a scary problem into a two‑step breeze. Always look for that hidden (x − k) lurking in your expression. Dividing Polynomials - Long Division Method
  7. Order by Degree - Always arrange both dividend and divisor in descending powers of x. This neat setup prevents misalignment and keeps your work tidy, kind of like sorting your music playlist before a jam session. A little organization goes a long way toward error‑free results. Dividing Polynomials - Long Division Method
  8. Avoid Common Pitfalls - Watch out for misaligned terms and the elusive forgotten remainder. Double‑check each subtraction step and make sure you haven't skipped a zero placeholder. Catching these small errors keeps you on track and feeling like a math champion. Dividing Polynomials - Long Division Method
  9. Know When Synthetic Works - Synthetic division only flies when your divisor is linear and monic (leading coefficient 1). If you see something like 2x − 3, you'll need to adapt or fall back on long division. Picking the right tool for the job is half the battle. Synthetic Division (Definition, Steps and Examples)
  10. Account for Missing Terms - If a polynomial skips a power (say x³ to x¹), pop in a 0x² to keep everything lined up. These invisible placeholders are your secret weapon for smooth, error‑free division. It's like putting in the missing puzzle piece so the picture comes together perfectly. Polynomials - Long Division
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