Which of the following is not a polynomial identity?
A. [tex]$a^3+b^3-(a-b)\left(a^2-a b+b^2\right)$[/tex]
B. [tex]$\left(a^2+b^2\right)\left(c^2+d^2\right)=(a c-b d)^2+(a d+b c)^2$[/tex]
C. [tex]$a^2-b^2=(a+b)(a-b)$[/tex]
D. [tex]$a^3-b^3-(a-b)\left(a^2+a b+b^2\right)$[/tex]
Answer (2)
After analyzing each option, Option A is not a polynomial identity as it does not equal zero for all values of a and b. Options B, C, and D are all valid polynomial identities. Therefore, the correct answer is A.
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Simplify each expression to check if it is an identity.
Option A simplifies to 2 b ( b 2 + a 2 − ab ) , which is not always zero.
Options B, C, and D are all polynomial identities.
Therefore, the expression that is not a polynomial identity is A .
Explanation
Understanding the Problem We are given four equations and asked to identify which one is not a polynomial identity. A polynomial identity is an equation that is true for all values of the variables. We need to check each equation to see if it holds true for all values of a, b, c, and d.
Analyzing Option A Let's analyze each option:
Option A: a 3 + b 3 − ( a − b ) ( a 2 − ab + b 2 ) . We need to simplify this expression and see if it equals zero (or some other constant) for all values of a and b .
Simplifying Option A Expanding and simplifying Option A:
a 3 + b 3 − ( a − b ) ( a 2 − ab + b 2 ) = a 3 + b 3 − ( a 3 − a 2 b + a b 2 − a 2 b + a b 2 − b 3 ) = a 3 + b 3 − a 3 + a 2 b − a b 2 + a 2 b − a b 2 + b 3 = 2 b 3 + 2 a 2 b − 2 a b 2 = 2 b ( b 2 + a 2 − ab ) .
Since 2 b ( b 2 + a 2 − ab ) is not always zero for all a and b , Option A is likely not an identity.
Analyzing Option B Option B: ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 2 . We need to simplify both sides and see if they are equal.
Simplifying Option B Expanding and simplifying Option B:
Left side: ( a 2 + b 2 ) ( c 2 + d 2 ) = a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 .
Right side: ( a c − b d ) 2 + ( a d + b c ) 2 = a 2 c 2 − 2 a c b d + b 2 d 2 + a 2 d 2 + 2 a d b c + b 2 c 2 = a 2 c 2 + b 2 d 2 + a 2 d 2 + b 2 c 2 .
Since both sides are equal, Option B is an identity.
Analyzing Option C Option C: a 2 − b 2 = ( a + b ) ( a − b ) . This is the difference of squares identity.
Simplifying Option C Expanding and simplifying Option C:
( a + b ) ( a − b ) = a 2 − b 2 . This is a well-known identity.
Analyzing Option D Option D: a 3 − b 3 − ( a − b ) ( a 2 + ab + b 2 ) . We need to simplify this expression.
Simplifying Option D Expanding and simplifying Option D:
a 3 − b 3 − ( a − b ) ( a 2 + ab + b 2 ) = a 3 − b 3 − ( a 3 + a 2 b + a b 2 − a 2 b − a b 2 − b 3 ) = a 3 − b 3 − a 3 − a 2 b − a b 2 + a 2 b + a b 2 + b 3 = 0 .
Since the expression simplifies to 0, Option D is an identity.
Conclusion Since Option A simplifies to 2 b ( b 2 + a 2 − ab ) , which is not always zero, Option A is not an identity. Options B, C, and D are all polynomial identities.
Examples
Polynomial identities are useful in simplifying algebraic expressions and solving equations. For example, the difference of squares identity, a 2 − b 2 = ( a + b ) ( a − b ) , can be used to quickly factor expressions. In engineering, polynomial identities can help simplify complex formulas, making calculations easier and more efficient. For instance, when designing structures, engineers use polynomial identities to analyze stress and strain distributions, ensuring the stability and safety of the design. Similarly, in computer graphics, polynomial identities are used to optimize rendering algorithms, leading to faster and more realistic image generation.
