Question

question 3 of 10
which of the following is not a polynomial identity?

a. ( a^2 - b^2 = (a + b)(a - b) )

b. ( a^3 + b^3 = (a - b)(a^2 - ab + b^2) )

c. ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) )

d. ( (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad + bc)^2 )

Explanation:

Step1: Check Option A

$a^2 - b^2 = (a+b)(a-b)$ is the difference of squares identity, valid.

Step2: Check Option B

Expand RHS: $(a-b)(a^2 - ab + b^2) = a^3 - a^2b + ab^2 - a^2b + ab^2 - b^3 = a^3 - 2a^2b + 2ab^2 - b^3 ≠ a^3 + b^3$. Invalid.

Step3: Check Option C

$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ is the difference of cubes identity, valid.

Step4: Check Option D

Expand RHS: $(ac+bd)^2 + (ad+bc)^2 = a^2c^2 + 2acbd + b^2d^2 + a^2d^2 + 2adbc + b^2c^2 = a^2c^2 + a^2d^2 + b^2d^2 + b^2c^2 = (a^2 + b^2)(c^2 + d^2)$. Valid.

Answer:

B. $a^3 + b^3 = (a - b)(a^2 - ab + b^2)$