Which of the following is not a polynomial identity?

a. (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc )
b. (a+b)^2=a^2+2ab+b^2
c. a^3-b^3=(a-b)(a^2+ab+b^2)
d. a^2(b+c)=a^3(b/a+c),

A polynomial identity is the identity whose left terms gives exactly right side terms.

In order to check if which of those have left side equals right side, we need to expand both side to the simplest step.

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

We could foil left side (a^2+b^2)(c^2+d^2)

It gives a^2c^2+a^2d^2+b^2c^2+b^2d^2.

Let us expand right side (ac-bd)^2+(ad+bc)

We get (ac-bd)^2 = a^2c^2+b^2d^2-2acbd

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

Left side a^2c^2+a^2d^2+b^2c^2+b^2d^2 is not equal to right side a^2c^2+b^2d^2-2acbd +ad+bc.

Therefore, A. (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc) is not a polynomial identity.

B. (a+b)^2=a^2+2ab+b^2

Let us expand left side of the identity, we get

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

We got left side a^2 +2ab + b^2 equals right side a^2+2ab+b^2.

Therefore, B. (a+b)^2=a^2+2ab+b^2 is a polynomial identity.

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

Let us expand right side of the identity, we get

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

We got left side a^3-b^3 equals right side a^3-b^2.

Therefore, C. a^3-b^3=(a-b)(a^2+ab+b^2) is a polynomial identity.

D. a^2(b+c)=a^3(b/a +c)

Expanding left side

a^2(b+c) = a^2b + a^2c

Expanding right side

a^3(b/a +c) = a^3*b/a + a^3c = a^2b +a^3c.

Left side a^2b + a^2c is not equal to right side a^2b +a^3c.

Therefore, D. a^2(b+c)=a^3(b/a +c) is not a polynomial identity.

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