Polynomial Identities Applications Quiz � Real-World Problems & Cubes (HSA.APR.C.4)

1) Which expression simplifies to a difference of squares?

(2x + 5)(2x − 5)

(x + 3)(x + 3)

(x − 2)²

(3x + 4)(3x + 4)

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

Explanation

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

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?

X³ + 3x² + 3x + 3

X³ + 9x² + 27x + 27

X³ + 6x² + 9x + 3

(x³ + 3)³

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

Explanation

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

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

(a + b)(a − b) = a² − b²

(a − b)² = a² − 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)² = a² + 2ab + b²

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

Explanation

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

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?

2500 + 300 + 9

2500 + 30 + 9

2500 + 2(50)(3) + 9

2500 + 150 + 9

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

Explanation

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

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:

2x² + 4

2x² + 8x + 8

2x² − 8

2x² + 8

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

Explanation

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

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

(x² + 4 − 2x)(x² + 4 + 2x)

X⁴ + 8x² + 16 − 4x²

X⁴ + 4x² + 8x + 16

(x − 2)⁴

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

Explanation

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

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

X³ + 3x² + 6x + 8

X³ + 4x² + 4x + 8

X³ + 6x² + 12x + 8

X³ + 8x² + 12x + 6

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

Explanation

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

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

It becomes x⁴ − 4x² + 4

It remains x² − 4

It becomes (x² − 4)²

It becomes x⁴ + 4

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

Explanation

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

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

98² = (100 − 2)²

98 × 98 = 98²

X² + y² = z²

7x + 3y

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

Explanation

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

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

X² + 3x

X² + 9

X² + 6x + 9

2x² + 3

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

Explanation

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

Real-World and Advanced Applications of Polynomial Identities

1) Which expression simplifies to a difference of squares?

2)

What first name or nickname would you like us to use?

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?

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

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?

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:

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

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

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

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

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