Real-World and Advanced Applications of Polynomial Identities

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7288 | Total Attempts: 9,526,515
| Questions: 10 | Updated: Nov 12, 2025
Please wait...
Question 1 / 10
0 %
0/100
Score 0/100
1) Which expression simplifies to a difference of squares?

Explanation

This matches the pattern (a + b)(a − b) = a² − b², giving 4x² − 25.

Submit
Please wait...
About This Quiz
Real-world And Advanced Applications Of Polynomial Identities - Quiz

Polynomial identities aren’t just abstract math—they appear in real-world applications and advanced problems. In this quiz, you’ll push your understanding further, applying identities in complex contexts. Try this quiz to see how algebra connects beyond the classroom.

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) The volume of a cube increases when each edge is lengthened by 3 units. If the original edge is x, what polynomial represents the new volume?

Explanation

New edge = x + 3. Volume = (x + 3)³. Expanding gives x³ + 9x² + 27x + 27.

Submit
3) Which polynomial identity is most useful for quickly calculating 105²?

Explanation

105² can be computed as (100 + 5)², which fits (a + b)².

Submit
4) A landscaper uses the identity (a + b)² to calculate the area of a square with side length (50 + 3) m. What is the area?

Explanation

(a + b)² with a = 50, b = 3 gives a² + 2ab + b² = 2500 + 300 + 9.

Submit
5) A contractor designs two square patios, one with side length x + 2 and another with side x − 2. The combined area of the patios is:

Explanation

(x + 2)² + (x − 2)² = (x² + 4x + 4) + (x² − 4x + 4) = 2x² + 8.

Submit
6) Which expression is equivalent to (x² + 4)² − (2x)²?

Explanation

This matches the difference of squares identity: A² − B² = (A − B)(A + B), where A = x² + 4 and B = 2x.

Submit
7) Using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³, expand (x + 2)³.

Explanation

Substituting a = x and b = 2 gives x³ + 3x²(2) + 3x(4) + 8.

Submit
8) If (x + 2)(x − 2) = x² − 4, what happens to the result if both terms are squared before multiplying?

Explanation

Squaring both factors gives (x + 2)²(x − 2)² = [(x + 2)(x − 2)]² = (x² − 4)².

Submit
9) Which expression shows the use of a polynomial identity to describe a numerical relationship?

Explanation

This uses (a − b)² = a² − 2ab + b² to calculate 98² quickly.

Submit
10) A square garden has side length (x + 3) meters. Which expression represents its area?

Explanation

The area is (x + 3)². Expanding using (a + b)² = a² + 2ab + b² gives x² + 6x + 9.

Submit
×
Saved
Thank you for your feedback!
10)
Your input helps us improve, and you’ll get your detailed results next.
View My Results
Cancel
  • All
    All (10)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Which expression simplifies to a difference of squares?
The volume of a cube increases when each edge is lengthened by 3...
Which polynomial identity is most useful for quickly calculating...
A landscaper uses the identity (a + b)² to calculate the area of...
A contractor designs two square patios, one with side length x + 2 and...
Which expression is equivalent to (x² + 4)² − (2x)²?
Using the identity (a + b)³ = a³ + 3a²b + 3ab² + b³, expand (x +...
If (x + 2)(x − 2) = x² − 4, what happens to the result if both...
Which expression shows the use of a polynomial identity to describe a...
A square garden has side length (x + 3) meters. Which expression...
Alert!

Advertisement