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Quizzes > High School Quizzes > Mathematics

Synthetic & Polynomial Long Division Practice Quiz

Build confidence in synthetic and polynomial division

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting Synthetic Poly Showdown, a chemistry trivia for high school and early college students.

What is synthetic division used for?
Multiplying two polynomials
Finding the derivative of a polynomial
Factoring a quadratic equation
Dividing a polynomial by a binomial of the form (x - a)
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - a). This method simplifies the long division process by reducing the arithmetic involved.
In synthetic division, when dividing by (x - a), which value is used as the divisor?
a
-a
x
None of the above
When dividing by (x - a), the value a is used because the process is based on evaluating the polynomial at x = a. This is also the key concept behind the Remainder Theorem.
What must be done if a term is missing in a polynomial before applying synthetic division?
Factor the polynomial first
Ignore the missing term
Insert a 0 for the missing term
Reorder the terms
Before performing synthetic division, it is important to include a 0 coefficient for any missing term so that every degree is represented. This ensures the columns line up correctly during the division process.
What does the Remainder Theorem state regarding polynomial division?
It only applies to quadratic polynomials
The remainder is always zero
The remainder of f(x) divided by (x - a) is f(a)
The quotient is always f(a)
The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a). This theorem provides a quick method to evaluate the remainder without performing full division.
In polynomial long division, what is the first step of the process?
Divide the leading term of the dividend by the leading term of the divisor
Subtract the constants
Factor both polynomials
Multiply the entire dividend by 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 and sets up the subsequent subtraction steps.
Using synthetic division, divide f(x) = 2x³ + 3x² - x - 6 by (x - 2). What is the remainder?
6
14
0
20
By setting up synthetic division with 2, the coefficients 2, 3, -1, and -6 lead to a final remainder of 20. This confirms that f(2) is 20 as predicted by the Remainder Theorem.
Perform synthetic division on f(x) = x³ - 6x² + 11x - 6 using (x - 1). What is the resulting quotient?
x² - 6x + 11
x² - 5x + 6
x² + 5x - 6
x² + 6x + 6
Using synthetic division with the value 1 gives quotient coefficients of 1, -5, and 6, corresponding to the polynomial x² - 5x + 6. This process confirms that (x - 1) divides f(x) correctly.
When dividing f(x) = 3x³ + 0x² - 2x + 4 by (x + 2) using synthetic division, which value should be used?
2
-2
0
Synthetic division cannot be used
Since the divisor is (x + 2), it can be rewritten as (x - (-2)). Therefore, the value -2 is used in synthetic division. This change aligns with the technique required by the Remainder Theorem.
Using polynomial long division, find the quotient when dividing f(x) = 4x³ + 8x² - 3x + 5 by (2x + 1).
2x² + 3x - 1
2x² + 3x + 1
2x² + 3x - 3
2x² + 3x + 3
Dividing the leading terms and subtracting the intermediate products step-by-step reveals that the quotient is 2x² + 3x - 3. The remainder is present, but the question solely asks for the quotient.
Apply the Remainder Theorem to f(x) = x❴ - 5x³ + 4x² + 2x - 8 using the divisor (x - 2). What is the remainder?
-12
-8
0
12
Evaluating f(2) gives 16 - 40 + 16 + 4 - 8 = -12. According to the Remainder Theorem, this result is the remainder when f(x) is divided by (x - 2).
In synthetic division of f(x) = 5x³ - 4x² + 3x - 2 by (x - 1), what is the number obtained after the first multiplication and addition step?
3
1
-4
5
After bringing down the first coefficient (5) and multiplying it by 1, adding to -4 yields 1. This calculation is the essential first step in the synthetic division process.
What is a necessary condition for using synthetic division on a polynomial?
The coefficients must be integers
The polynomial must be quadratic
The divisor must be linear (of the form x - c)
The dividend must be of higher degree than the divisor
Synthetic division is only applicable when the divisor is a linear expression of the form (x - c). This limitation is due to the structure of the method and its reliance on the Remainder Theorem.
In polynomial long division, what happens if the degree of the dividend is lower than the degree of the divisor?
The quotient is 0 and the dividend becomes the remainder
The dividend is multiplied by x until degrees match
The division cannot be performed
The divisor is divided by the dividend
If the degree of the dividend is lower than that of the divisor, no division occurs and the quotient is 0. In this case, the entire dividend is treated as the remainder.
Perform synthetic division on f(x) = x³ + 2x² - 5x + 6 by (x - 2). What is the quotient polynomial?
x² + 4x + 3
x² + 4x - 3
x² + 2x - 5
x² + 2x + 6
Synthetic division with 2 applied to the coefficients of f(x) produces new coefficients corresponding to the quotient polynomial x² + 4x + 3. This process confirms correct alignment and computation.
Which technique is generally more efficient for dividing a polynomial by a binomial of the form (x - c)?
Factoring
Synthetic division
Polynomial long division
Substitution
Synthetic division offers a streamlined process compared to polynomial long division when the divisor is linear. Its efficiency makes it a preferred method in these cases.
Divide f(x) = 2x❴ - 3x³ + x² - 5x + 4 by (x² - x + 1) using polynomial long division. What is the quotient?
2x² - x - 2
2x² - x - 2 remainder -6x + 6
x² - 2x + 2
2x² - x + 2
By dividing 2x❴ by x² and proceeding with the subtraction steps, the quotient is determined to be 2x² - x - 2. The process involves careful alignment of polynomial terms, and the remainder is not required for this answer.
Using synthetic division, compute f(3) for f(x) = 4x³ - 2x² + x - 7. What is the value and its significance?
3
0
-86
86
Performing synthetic division with 3 gives a final value of 86, which is f(3) per the Remainder Theorem. This value represents the remainder when the polynomial is divided by (x - 3).
In synthetic division, if the final remainder is zero, what can you conclude about the polynomial and the divisor (x - c)?
The divisor is a repeated factor
There must be an error, as a zero remainder is impossible
x = c is a root of the polynomial, and (x - c) is a factor
The polynomial does not factor over the reals
A remainder of zero indicates that f(c) = 0. According to the Factor Theorem, this means that x = c is a root and that (x - c) is a factor of the polynomial.
Why is it important to write both the dividend and divisor in descending order, including zeros for missing terms, in polynomial long division?
It makes the division process faster
It is only necessary for synthetic division
It ensures accurate alignment of like terms during subtraction
It automatically factors the polynomial
Writing polynomials in descending order with placeholders for missing terms ensures that like terms align correctly. This precise alignment is crucial for accurately subtracting and simplifying terms during long division.
Consider f(x) = 6x❴ + ax³ + bx² + cx + d. If synthetic division by (x - 2) produces a remainder of zero, which equation is implied by the Remainder Theorem?
a(16) + b(8) + c(4) + d(2) = 0
6(16) + 8a + 4b + 2c + d = 0
6(16) + a + b + c + d = 0
16a + 8b + 4c + 2d = 0
According to the Remainder Theorem, if (x - 2) is a factor then f(2) = 0. Substituting x = 2 into f(x) gives 6(16) + a(8) + b(4) + c(2) + d = 0, which is the equation stated in the correct answer.
0
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Study Outcomes

  1. Analyze synthetic pathways to predict reaction products.
  2. Apply principles of polymer analysis in evaluating chemical structures.
  3. Demonstrate proficiency in synthetic organic chemistry problem-solving.
  4. Execute polynomial long division to simplify complex reaction equations.
  5. Synthesize key concepts from reaction mechanisms and polymer behavior.

4.04 Quiz: Synthetic & Polynomial Division Cheat Sheet

  1. Retrosynthetic Analysis - Ever felt overwhelmed by a mega molecule? Break it down step-by-step into simpler building blocks to plan your perfect synthesis route. This detective-like approach turns complex targets into a clear roadmap for success. Synthesis Road Map Problems in Organic Chemistry
  2. Key Synthetic Techniques - Master alkane halogenation, organometallic alkylation, and essential functional group transformations to become a reaction powerhouse. These core methods are your toolkit for building and tweaking organic compounds with precision. Synthetic Cheatsheet Explained
  3. Reaction Mechanisms - Dive into nucleophilic substitution, elimination, and addition reactions to predict how molecules will behave. Understanding each step lets you map out and control your synthetic pathways like a pro. Organic Chemistry: Synthetic Pathways
  4. Conditions & Reagents - Learn when to use UV light for radical substitutions or heat under reflux for nucleophilic swaps. Picking the right conditions and reagents is the secret sauce for clean, high-yield reactions. Organic Chemistry: Synthetic Pathways
  5. Multi‑step Route Design - Practice sketching out multi-step syntheses where each intermediate is feasible and efficient. This skill transforms you into a molecular architect, capable of tackling even the most complex targets. Synthesis Road Map Problems in Organic Chemistry
  6. Polymerization Principles - Get the lowdown on addition and condensation polymerization to craft polymers with tailored properties. Knowing how monomers link up lets you engineer plastics, fibers, and gels for any application. Organic Chemistry: Synthetic Pathways
  7. Gel Permeation Chromatography - Discover how GPC helps you analyze polymer molecular weight distributions like a boss. This powerhouse technique is key for quality control and material design in polymer chemistry. Gel Permeation Chromatography
  8. Protecting Groups - Use protecting groups to shield sensitive functional groups and steer reactions exactly where you want them. This clever strategy avoids unwanted side-reactions and boosts overall yields. Synthesis Road Map Problems in Organic Chemistry
  9. Stereochemistry - Keep an eye on the 3D arrangement of atoms to ensure your molecules have the right biological activity. Mastering stereochemistry is crucial for drug design and other bioactive compounds. Organic Chemistry: Synthetic Pathways
  10. Flashcards & Quizzes - Reinforce your skills with fun quizzes and flashcard decks that test your synthesis knowledge on the go. Regular practice is the fastest route to confidence and mastery in organic synthesis. Synthetic Pathways Flashcards
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