Question

Simplify the following: a

2

(b2 – c

2

) + b2

(c2 – a

2

) + c2

(a2 – b

2

)​

Answer 1

$\boxed{a^{2}(b^{2}-c^{2})+b^{2}(c^{2}-a^{2})+c^{2}(a^{2}-b^{2})=0}$

Step-by-step explanation:

Now, $a^{2}(b^{2}-c^{2})+b^{2}(c^{2}-a^{2})+c^{2}(a^{2}-b^{2})$

• Hint: Let us multiply the terms using the algebraic operation:
• $\quad a(b+c)=ab+ac$
• or, $a(b-c)=ab-ac$

$=a^{2}b^{2}-a^{2}c^{2}+b^{2}c^{2}-b^{2}a^{2}+c^{2}a^{2}-c^{2}b^{2}$

• Hint: Now, we rewrite the terms using commutative property of multiplication:
• $\quad ab=ba$

$=a^{2}b^{2}-c^{2}a^{2}+b^{2}c^{2}-a^{2}b^{2}+c^{2}a^{2}-b^{2}c^{2}$

• Hint: We rearrange the terms.

$=a^{2}b^{2}-a^{2}b^{2}+b^{2}c^{2}-b^{2}c^{2}+c^{2}a^{2}-c^{2}a^{2}$

• Hint: Since the difference of similar terms is $0$, we write:
• $\quad a^{2}b^{2}-a^{2}b^{2}=0$

$=0+0+0$

$=\bold{0}$

Answer 2

Answer:

Step-by-step explanation:

Now, a^{2}(b^{2}-c^{2})+b^{2}(c^{2}-a^{2})+c^{2}(a^{2}-b^{2})a

2

(b

2

−c

2

)+b

2

(c

2

−a

2

)+c

2

(a

2

−b

2

)

Hint: Let us multiply the terms using the algebraic operation:

\quad a(b+c)=ab+aca(b+c)=ab+ac

or, a(b-c)=ab-aca(b−c)=ab−ac

=a^{2}b^{2}-a^{2}c^{2}+b^{2}c^{2}-b^{2}a^{2}+c^{2}a^{2}-c^{2}b^{2}=a

2

b

2

−a

2

c

2

+b

2

c

2

−b

2

a

2

+c

2

a

2

−c

2

b

2

Hint: Now, we rewrite the terms using commutative property of multiplication:

\quad ab=baab=ba

=a^{2}b^{2}-c^{2}a^{2}+b^{2}c^{2}-a^{2}b^{2}+c^{2}a^{2}-b^{2}c^{2}=a

2

b

2

−c

2

a

2

+b

2

c

2

−a

2

b

2

+c

2

a

2

−b

2

c

2

Hint: We rearrange the terms.

=a^{2}b^{2}-a^{2}b^{2}+b^{2}c^{2}-b^{2}c^{2}+c^{2}a^{2}-c^{2}a^{2}=a

2

b

2

−a

2

b

2

+b

2

c

2

−b

2

c

2

+c

2

a

2

−c

2

a

2

Hint: Since the difference of similar terms is 00 , we write:

\quad a^{2}b^{2}-a^{2}b^{2}=0a

2

b

2

−a

2

b

2

=0

=0+0+0=0+0+0

=\bold{0}=0