match each polynomial on the left with its two factors on the right

Question

8x^3+1
8x^3-1
2x-1
2x+1
4x^2+4x+1
4x^2+2x+1
4x^2-4x-1
4x^2-2x+1

Answer 1

To match the given polynomials on the left with their factors on the right, let's factor the polynomials on the left and see which pairs match up with the factors provided.

1. Polynomial: \( 8x^3 + 1 \)
   This can be factored using the sum of cubes formula:
   \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
   Here, \( a = 2x \) and \( b = 1 \):
   \[ 8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1) \]

Match: \( 8x^3 + 1 \) with \( 2x + 1 \) and \( 4x^2 - 2x + 1 \)

2. Polynomial: \( 8x^3 - 1 \)
   This can be factored using the difference of cubes formula:
   \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
   Here, \( a = 2x \) and \( b = 1 \):
   \[ 8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1) \]

Match: \( 8x^3 - 1 \) with \( 2x - 1 \) and \( 4x^2 + 2x + 1 \)

Summary of Matches:

• \( 8x^3 + 1 \) matches with \( 2x + 1 \) and \( 4x^2 - 2x + 1 \).
• \( 8x^3 - 1 \) matches with \( 2x - 1 \) and \( 4x^2 + 2x + 1 \).

List of Matches:

1. \( 8x^3 + 1 \) → \( 2x + 1 \), \( 4x^2 - 2x + 1 \)
2. \( 8x^3 - 1 \) → \( 2x - 1 \), \( 4x^2 + 2x + 1 \)

The other polynomials \( 4x^2 + 4x + 1 \) and \( 4x^2 - 4x - 1 \) do not correspond to any cubic polynomials mentioned above for factoring.