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Quizzes > Mathematics

Polynomial Test: Practice Problems to Sharpen Your Skills

Quick, free polynomial quiz with 20 problems. Instant feedback and answers.

Editorial: Review CompletedCreated By: Matt BolandUpdated Aug 23, 2025
Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting Polynomial Power Play, a high school math quiz.

This quiz helps you check your skills with polynomials, from basic operations to division. Work through 20 short questions, see instant answers, and spot topics to review before a test. For more algebra practice, try the solve for x quiz or the algebra 1 quiz.

What is the degree of the polynomial 3x^2 - 2x + 7?
2
0
3
1
The degree of a polynomial is determined by its highest exponent. Since the highest exponent in 3x^2 - 2x + 7 is 2, the degree is 2.
Simplify the expression: (2x + 3) + (4x - 5).
2x - 2
6x + 8
6x - 2
6x - 8
By combining like terms, 2x + 4x yields 6x and 3 - 5 gives -2. This results in the simplified expression 6x - 2.
In the polynomial 5x^3 + 2x^2 - 7x + 9, what is the constant term?
5
9
2
-7
The constant term is the term that does not contain any variables. In this polynomial, 9 is the term without an x attached.
Multiply 3 by the polynomial 2x^2 - x + 4.
3x^2 - x + 4
6x^2 + 3x + 12
6x^2 - 3x + 12
2x^2 - x + 4
Multiplying each term of 2x^2 - x + 4 by 3 produces 6x^2, -3x, and 12 respectively. This demonstrates the distributive property in scalar multiplication.
In the polynomial 6x^2 + 3x - 5, what is the coefficient of x?
3
6
-5
x
The coefficient is the numerical part of the term that contains the variable. In 3x, the number 3 is the coefficient of x.
Simplify: 4x^2 + 3x - 5 + 2x^2 - 4x + 7.
6x^2 + x + 2
6x^2 - 7x + 2
2x^2 - x + 2
6x^2 - x + 2
Combine like terms: adding 4x^2 and 2x^2 gives 6x^2, 3x and -4x combine to -x, and -5 with 7 results in 2. Thus, the simplified form is 6x^2 - x + 2.
What is the result of adding (3x^2 - 2x + 1) and (2x^2 + 5x - 3)?
5x^2 + 3x - 2
6x^2 + 3x - 2
5x^2 + 3x + 2
5x^2 + 7x - 2
By adding the corresponding coefficients, 3x^2 and 2x^2 add up to 5x^2, -2x and 5x add to 3x, and 1 plus (-3) gives -2. This yields the polynomial 5x^2 + 3x - 2.
Find the result of subtracting the polynomial (x^2 + 4x - 6) from (3x^2 - x + 2).
x^2 - 5x + 8
4x^2 + 3x - 4
2x^2 - 5x + 8
2x^2 + 5x + 8
Subtract the corresponding terms of the two polynomials: 3x^2 - x^2 gives 2x^2, -x - 4x is -5x, and 2 - (-6) results in 8. The final expression is 2x^2 - 5x + 8.
Evaluate f(x) = 2x^2 - 3x + 4 when x = 2.
8
4
10
6
Substitute x = 2 into the polynomial: 2(2)^2 - 3(2) + 4 equals 8 - 6 + 4, which simplifies to 6. This is a straightforward evaluation of the function.
Factor the polynomial 4x^2 + 8x.
2x(2x+8)
4x(x+2)
x(4x+8)
4x(x+8)
Both terms in the polynomial have a common factor of 4x. Factoring 4x out of 4x^2 + 8x results in 4x(x+2).
Factor the quadratic x^2 + 5x + 6.
(x+1)(x+6)
(x+2)(x+4)
(x+2)(x+3)
(x+3)(x+4)
Look for two numbers that multiply to 6 and add up to 5; these are 2 and 3. Therefore, the quadratic factors nicely into (x+2)(x+3).
Factor the polynomial x^3 + 3x^2 + x + 3 by grouping.
(x^2+3x)(x+1)
(x+3)(x+1)
(x+3)(x^2+1)
(x+1)(x^2+3)
Group the terms as (x^3 + 3x^2) and (x + 3). Factoring each group shows a common factor of (x+3), resulting in (x+3)(x^2+1).
For the polynomial f(x) = -2x^3 + 5x^2 - x + 4, what is the end behavior as x approaches infinity?
f(x) approaches positive infinity
f(x) oscillates
f(x) approaches 0
f(x) approaches negative infinity
The term -2x^3 controls the end behavior because it is of the highest degree. Since the coefficient is negative and the degree is odd, as x approaches infinity, f(x) decreases without bound.
Simplify the expression: 3(x + 2) + 4(2x - 3).
7x - 6
11x - 6
7x + 6
11x + 6
First, distribute to get 3x + 6 and 8x - 12. Adding these gives 11x and -6 for the constant term, resulting in 11x - 6.
What is the leading coefficient of the polynomial -7x^4 + 3x^3 - x + 5?
3
-7
1
5
The leading coefficient is the coefficient of the term with the highest power of x. In this case, the term -7x^4 has the highest power, so the leading coefficient is -7.
Divide the polynomial 2x^3 - 3x^2 + 4x - 5 by x - 1.
2x^2 - x + 3 with remainder 2
2x^2 + x + 3 with remainder -2
2x^2 - x + 3 with remainder 0
2x^2 - x + 3 with remainder -2
Using polynomial long division, the quotient of 2x^3 - 3x^2 + 4x - 5 by x - 1 is 2x^2 - x + 3 with a remainder of -2. This result can also be verified using synthetic division.
Using the Remainder Theorem, find the remainder when f(x) = x^3 - 4x^2 + 5x - 2 is divided by x - 2.
0
-2
2
4
The Remainder Theorem states that the remainder of dividing f(x) by x - c is f(c). Evaluating f(2) gives 0, indicating that the remainder is 0.
Given f(x) = x^3 - 6x^2 + 11x - 6, which value of x is a root according to the Factor Theorem?
1
3
6
2
According to the Factor Theorem, if f(c) = 0 then (x - c) is a factor of f(x). Testing x = 1 yields f(1) = 0, which confirms that 1 is a root.
Multiply the polynomials: (x + 4)(x^2 - 4x + 7).
x^3 - 9x + 28
x^3 + 9x - 28
x^3 - 9x - 28
x^3 + 9x + 28
Multiplying (x + 4) by (x^2 - 4x + 7) results in x^3 - 4x^2 + 7x + 4x^2 - 16x + 28. Combining like terms, the x^2 terms cancel, and the x terms combine to -9x, giving x^3 - 9x + 28.
Find the coefficient of x^2 in the product (2x - 3)(x^2 + 4x + 5).
-3
2
8
5
When multiplying (2x - 3)(x^2 + 4x + 5), the x^2 terms arise from 2x·4x, which gives 8x^2, and -3·x^2, which subtracts 3x^2. Combining these yields 5x^2, so the coefficient of x^2 is 5.
0
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Study Outcomes

  1. Analyze polynomial expressions to identify their key components.
  2. Factor and simplify various polynomial problems accurately.
  3. Apply division techniques to solve complex polynomial equations.
  4. Solve polynomial equations and determine their roots.
  5. Graph polynomial functions to interpret their behavior in real-world contexts.

Polynomial Practice Problems Cheat Sheet

  1. Understand Polynomial Terminology - Start by getting comfy with the lingo: polynomials are sums of terms that mix variables and coefficients through addition, subtraction, and multiplication. You'll classify them as monomials, binomials, or trinomials based on how many terms they contain - and spot their degree by the highest exponent. Mastering these basics is your first step to turning complex expressions into clear, manageable puzzles. Explore Key Concepts
  2. Master Addition and Subtraction of Polynomials - Combine like terms like a pro by matching variable parts and adding or subtracting their coefficients. For instance, (3x² + 2x) + (5x² − 4x) becomes 8x² − 2x by simply pairing 3x² with 5x² and 2x with −4x. This skill makes simplifying any polynomial a breeze and lays the foundation for more advanced moves. Practice Simplification
  3. Apply the FOIL Method for Multiplying Binomials - FOIL stands for First, Outer, Inner, Last - a simple checklist to multiply (x + a)(x + b) without dropping any terms. For example, (x + 3)(x − 2) becomes x² − 2x + 3x − 6, which simplifies to x² + x − 6. This reliable trick turns binomial multiplication from a headache into a quick, four-step dance. Learn FOIL
  4. Recognize Special Products - Spot patterns like difference of squares (a² − b² = (a + b)(a − b)) and perfect square trinomials (a² + 2ab + b² = (a + b)²) to save time on multiplication and factoring. These shortcuts are math's secret handshake, revealing factors at a glance. Once you memorize a few key patterns, you'll feel like you've unlocked cheat codes for polynomial problems. View Special Products
  5. Understand the Remainder and Factor Theorems - The Remainder Theorem says plugging x = a into f(x) gives the remainder of f(x) ÷ (x − a), and the Factor Theorem adds that if f(a)=0, then (x − a) is a factor. These two theorems are like a two-step diagnostic for divisibility and roots. They make checking factors as easy as evaluating a function at one point. Check Theorems
  6. Practice Factoring Techniques - Always start by pulling out the greatest common factor (GCF). Then, for trinomials like x² + 5x + 6, find two numbers that multiply to 6 and sum to 5 - they're 2 and 3 - so it factors to (x + 2)(x + 3). Regular practice turns factoring from guesswork into a methodical, satisfying process. Review Factoring Tips
  7. Explore Polynomial Graphs - The degree and leading coefficient determine how the ends of your graph wave - up, down, or one up and one down. For example, an even-degree polynomial with a positive leading coefficient has both ends pointing skyward. Visualizing these behaviors helps you sketch or predict graphs with confidence. See Graph Behaviors
  8. Learn Polynomial Division Methods - Use long division for any divisor and synthetic division when you're dividing by (x − a). Synthetic division condenses the process into a quick table of numbers, making it faster and less error‑prone. Mastering both tools broadens your toolkit for breaking down complex expressions. Division Techniques
  9. Understand the Fundamental Theorem of Algebra - This powerhouse theorem guarantees that every non‑zero polynomial of degree n has exactly n complex roots (counting multiplicity). It ensures you're never hunting for more roots than actually exist. Embracing this theorem gives you big-picture confidence in solving any polynomial equation. Discover the Theorem
  10. Apply Polynomials to Real-World Problems - From modeling projectile motion to calculating areas and profits, polynomials pop up everywhere in science and business. Translating word problems into polynomial expressions turns real-world puzzles into math you can solve step by step. Practice with varied applications to see how theory meets everyday life. Watch the Guide
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