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Quizzes > High School Quizzes > Electives

Synthetic Division Practice Quiz

Sharpen your remainder theorem and division skills

Difficulty: Moderate
Grade: Other
Study OutcomesCheat Sheet
Paper art illustrating a trivia quiz on synthetic division and remainders for high school students

What is synthetic division primarily used for in polynomial operations?
Dividing a polynomial by a binomial of the form (x - c)
Solving linear equations
Factoring quadratic equations
Multiplying two polynomials
Synthetic division is a simplified method used to divide a polynomial by a binomial of the form (x - c). It streamlines the long division process by focusing only on the coefficients.
According to the Remainder Theorem, what is the remainder when a polynomial f(x) is divided by (x - c)?
0
c
f(c)
x - c
The Remainder Theorem states that the remainder of the division of f(x) by (x - c) is f(c). This allows quick evaluation of the remainder without performing full division.
Before performing synthetic division on a polynomial, how should the polynomial be arranged?
Random order of the coefficients
In ascending order of the powers of x
In descending order of the powers of x, filling in any missing terms with 0
Grouped by even and odd powers
Polynomials must be arranged in descending order with placeholders (0s) for any missing degrees to correctly perform synthetic division. This arrangement ensures that each coefficient lines up with the appropriate power of x.
What is the main advantage of using synthetic division over long division for polynomials?
It works for dividing by any polynomial
It eliminates the need for the remainder theorem
It is faster and uses fewer calculations when dividing by linear factors
It is more accurate for high-degree polynomials
Synthetic division simplifies polynomial division by focusing on the coefficients, making it faster and more efficient for divisors of the form (x - c). This efficiency is particularly useful when handling linear factors.
Using the Remainder Theorem, how can you determine if (x - c) is a factor of f(x)?
By evaluating f(0) and checking if it equals c
By evaluating f(c) and checking if it equals 0
By performing synthetic division and analyzing the quotient
By finding the derivative of f(x)
The Factor Theorem, a direct consequence of the Remainder Theorem, states that if f(c) equals 0, then (x - c) is a factor of f(x). Evaluating f(c) is a quick method to test for factors.
Perform synthetic division on f(x) = 2x² + 3x - 5 divided by (x - 1). What is the remainder?
0
-5
1
5
Using synthetic division with c = 1, the process results in a remainder of 0. This indicates that (x - 1) divides the polynomial evenly.
Find the quotient when f(x) = x³ - 6x² + 11x - 6 is divided by (x - 2) using synthetic division.
x² + 4x - 3
x² - 4x + 3
x² - 2x + 1
x² - 6x + 11
Synthetic division transforms the coefficients to yield the quotient x² - 4x + 3 with a remainder of 0, confirming that 2 is a root of the polynomial.
Using synthetic division, determine f(4) for the polynomial f(x) = 3x³ + 2x² - x + 5.
217
225
230
221
By applying synthetic division with c = 4, the calculated value of f(4) is 225. This evaluation confirms the use of the Remainder Theorem in computing function values.
What is the remainder when f(x) = x³ + 4x² + 5 is divided by (x + 2)?
9
-13
13
0
According to the Remainder Theorem, substituting x = -2 into the polynomial yields f(-2) = (-8) + 16 + 5, which sums to 13. This value represents the remainder of the division.
For the polynomial f(x) = x³ + kx² + 6x - 8, if f(2) = 0, what is the value of k?
4
-4
3
-3
Substituting x = 2 into the polynomial gives 8 + 4k + 12 - 8, which simplifies to 4k + 12. Setting this equal to 0 leads to k = -3, confirming that 2 is a root.
When dividing f(x) = 4x³ - 2x² + x - 7 by (x + 1) using synthetic division, which value is used in the synthetic division box?
1
-1
7
0
For synthetic division, the divisor must be written in the form (x - c). Since (x + 1) can be expressed as (x - (-1)), the value used is -1.
Perform synthetic division on f(x) = 5x³ + 0x² - 3x + 4 divided by (x - 2). What is the coefficient of x in the quotient?
10
5
17
2
Synthetic division of the given polynomial yields a quotient with coefficients 5, 10, and 17. The coefficient corresponding to the x term is 10, determined through successive multiplication and addition.
Using the Remainder Theorem, find the remainder when f(x) = 2x❴ - 3x³ + x² - x + 6 is divided by (x - 2).
12
18
16
14
Evaluating f(2) using the Remainder Theorem, we substitute x = 2 into the polynomial which results in a remainder of 16. This method bypasses the need for long synthetic division.
For the polynomial f(x) = 3x³ + ax² + bx + c, synthetic division by (x - 1) gives a remainder of 2. What can be inferred about the sum a + b + c?
a + b + c = 2
a + b + c = 0
a + b + c = -1
a + b + c = 1
The Remainder Theorem states that f(1) is the remainder when dividing by (x - 1). Since f(1) equals 3 + a + b + c and is given as 2, rearranging yields a + b + c = -1.
If synthetic division of a polynomial f(x) by (x - 3) produces a remainder of 0, what does this indicate about (x - 3)?
(x - 3) is a factor of f(x)
(x - 3) has no impact on f(x)
(x - 3) is the quotient of f(x)
(x - 3) is not related to f(x)
A remainder of 0 means that (x - 3) divides f(x) evenly, which by the Factor Theorem indicates that (x - 3) is a factor of f(x).
Divide f(x) = 2x❴ - 3x³ + 4x² - 5x + 6 by (x + 2) using synthetic division. What are the quotient and remainder?
Quotient: 2x³ - 7x² + 18x - 41; Remainder: 41
Quotient: 2x³ - 7x² + 18x - 41; Remainder: -88
Quotient: 2x³ - 7x² + 18x - 41; Remainder: 88
Quotient: 2x³ - 7x² + 18x - 41; Remainder: 0
Performing synthetic division with c = -2 yields the sequence of coefficients that form the quotient 2x³ - 7x² + 18x - 41 and a remainder of 88. This method simplifies polynomial division by focusing on numerical computations.
For the polynomial f(x) = x❴ + p x³ + q x² + r x + s, synthetic division by (x - 1) yields a remainder of 0 and by (x + 2) yields a remainder of 20. What is f(-2)?
0
f(-2) cannot be determined from the information given
-20
20
The Remainder Theorem tells us that f(-2) is the remainder when f(x) is divided by (x + 2). Since the given remainder is 20, it directly follows that f(-2) = 20.
For the polynomial f(x) = 2x³ + kx² + 3x + m, if synthetic division by (x + 1) yields a remainder of 4, what is the relationship between k and m?
k - m = 9
k + m = -9
k + m = 9
k - m = -9
By the Remainder Theorem, substituting x = -1 into f(x) gives f(-1) = -2 + k - 3 + m which simplifies to k + m - 5. Setting this equal to the remainder 4 yields k + m = 9.
Using synthetic division, determine the remainder when f(x) = 6x❴ - x³ + 2x² - 3x + 5 is divided by (x - 1).
6
5
1
9
Evaluating f(1) using the Remainder Theorem (since the divisor is x - 1) yields 6 - 1 + 2 - 3 + 5, which sums to 9. This value is the remainder of the division.
If f(x) = x❵ - 2x❴ + 3x³ - 4x² + 5x - 6 is divided by (x - 2) using synthetic division, what is the remainder?
0
6
12
18
By the Remainder Theorem, evaluating f(2) gives the remainder when dividing by (x - 2). Substituting x = 2 into f(x) yields 12, which is confirmed by synthetic division.
0
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Study Outcomes

  1. Apply synthetic division to divide polynomials.
  2. Calculate remainders using the Remainder Theorem.
  3. Analyze the relationship between synthetic division and polynomial behavior.
  4. Verify solutions by checking consistency of remainders.
  5. Assess problem-solving approaches in synthetic division scenarios.

Synthetic Division & Remainder Theorem Cheat Sheet

  1. Synthetic division shortcut - Synthetic division zips through polynomial division by linear factors like x − c with way less scribbling than long division. Once you master the pattern, it feels like uncovering a brilliant math hack to slice through high‑degree polynomials in seconds. It's a snazzy trick that'll save you time and brainpower on every homework set! Symbolab Synthetic Division Guide
  2. Key steps of synthetic division - Start by listing the coefficients of your dividend, bring down the leading coefficient, multiply by c, add to the next coefficient, and repeat until you're out of numbers. This rhythmic process keeps you in the zone and prevents costly sign errors. Practice it a few times and soon you'll be clicking off problems like a pro! Symbolab Synthetic Division Guide
  3. Remainder theorem fundamentals - The Remainder Theorem says that dividing any polynomial f(x) by x − c leaves a remainder equal to f(c). In other words, plug in c and get your remainder without any actual division. It's a clever shortcut to check answers and spot roots in a snap! Symbolab Remainder & Factor Theorems
  4. Real‑world example practice - Try dividing x³ − 2x² + 4x − 3 by x − 2 using synthetic division and watch the magic unfold: your quotient becomes x² + 0x + 4 with a remainder of 5. Working through concrete examples like this builds confidence and cements the pattern in your mind. Plus, you'll love seeing the numbers align perfectly! Varsity Tutors Practical Example
  5. Quick remainder discovery - Forget full-on division: just evaluate f(c) to find the remainder when dividing by x − c. This direct plug‑and‑chug method cuts out extra steps and helps you check solutions instantly. It's like having a math cheat code for remainders! Symbolab Remainder & Factor Theorems
  6. When to use long division instead - Synthetic division only shines with linear divisors of the form x − c. If your divisor is anything more complex - like a quadratic or a binomial with variable terms on both sides - traditional long division is the better play. Knowing which tool to grab makes you a versatile problem‑solver! Online Math Learning Division Tutorial
  7. Avoid common coefficient slip‑ups - Always include zeros for missing degrees (e.g., x³ + 0x² − x + 5) or you'll end up with misaligned numbers and wrong answers. Double‑check your setup before you start bringing numbers down or multiplying. A little extra care up front saves a lot of head‑scratching later! Symbolab Synthetic Division Guide
  8. Rational root testing made easy - Wondering if a candidate c is a root? Run a quick synthetic division and see if your remainder is zero. A zero remainder means x − c is a factor and c is indeed a root. It's a speedy way to hunt down rational solutions! Symbolab Remainder & Factor Theorems
  9. Interpreting the quotient coefficients - The numbers you get after synthetic division form your quotient polynomial, which always has one degree less than the original. Treat those coefficients like building blocks for your new expression. They tell the next chapter of your polynomial's story! Symbolab Synthetic Division Guide
  10. Mix and match division techniques - Strengthen your skills by alternating between synthetic and long division problems. Some exercises may demand the precision of long division, while others let you breeze through with synthetic. Mastering both ensures you'll never be stumped by a polynomial problem again! Online Math Learning Division Tutorial
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