Question

Which of the following is not a polynomial identity?

Answer 1

$(a^{2} +b^{2} )(c^{2} -d^{2} )=(ac- b d)^{2} +(ad +bc)$ is not a polynomial identity.

What is polynomial identity?

"Polynomial identity is defined as the equation which helps us to solve the algebraic equation just by using it directly."

According to the question,

Verify given polynomial identity,

A. $a^{3} -b^{3} = (a-b)(a^{2} +ab + b^{2} )$

Right hand side

$= (a-b)(a^{2} +ab + b^{2} )\\=a(a^{2} +ab + b^{2} )-b(a^{2} +ab + b^{2} )\\=a^{3} +a^{2} b+ ab^{2} - ba^{2}-ab^{2}-b^{3}\\=a^{3}-b^{3}$

= Left hand side

This is a polynomial identity.

B. $(a+b)^{2} = a^{2} +2ab+b^{2}$

Left hand side

$(a +b)^{2}\\=(a +b)(a +b)\\=a(a + b) +b(a + b) \\= a^{2} +ab + b a + b^{2}\\ =a^{2} +2ab + b^{2}$

=Right hand side

This is a polynomial identity.

C. $(a^{2} +b^{2} )(c^{2} -d^{2} )=(ac- b d)^{2} +(ad +bc)$

Left hand side

$(a^{2} +b^{2} )(c^{2} -d^{2} )\\=a^{2}(c^{2} -d^{2} )+b^{2} (c^{2} -d^{2} )\\=a^{2} c^{2} -a^{2} d^{2} +b^{2} c^{2} -b^{2} d^{2}$                                     ____(1)

Right hand side

$=(ac- b d)^{2} +(ad +bc)\\=a^{2} c^{2} -2acbd +b^{2} d^{2} + ad +bc$                               ____(2)

From (1) and (2) we get,

Left hand side ≠ Right hand side

This is not a polynomial identity.

D. $a^{2} -b^{2} =(a +b)(a -b)$

Right hand side

$=(a +b)(a -b)\\=a(a-b) + b(a -b)\\=a^{2} -ab +ba-b^{2} \\=a^{2} -b^{2}$

=Left hand side

This is a polynomial identity.

Hence, Option(C) is not a polynomial identity.

Learn more about  polynomial identity here

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