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Using factor theorem, show that a-b, …

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Using factor theorem, show that a-b, b-c, c-a are the factors of a(b2-c2) +b(c2-a2) +c(a2-b2).

  • 1 answers

Rashmi Bajpayee 6 years, 8 months ago

If a - b is a factor of given expression, then a - b = 0 => a = b

Putting a = b, in the given expression, we get

b(b<font size="2">2</font>-c<font size="2">2</font>) +b(c<font size="2">2</font>- b<font size="2">2</font>) +c(b<font size="2">2</font>- b<font size="2">2</font>)

= b3 - bc<font size="2">2</font> + bc<font size="2">2</font>  - b<font size="2">3</font> + c(0)

= 0

Therefore, (a - b) is a factor of given expression.

Again if (b - c) is a factor of given expression, then

Putting b - c = 0 => b = c in the given expression, we get

a(c<font size="2">2 </font>- c<font size="2">2</font>) +c(c<font size="2">2 </font>- a<font size="2">2</font>) + c(a<font size="2">2 </font>- c<font size="2">2</font>)

= a(0) + c3 - ca2 + ca2 - c3 

= 0

Therefore, (b - c) is a factor of given expression.

Again if (c - a) is a factor of given expression, then

Putting c - a = 0 => c = a in the given expression, we get

a(b<font size="2">2 </font>- a<font size="2">2</font>) + b(a<font size="2">2 </font>- a<font size="2">2</font>) + a(a<font size="2">2 </font>- b<font size="2">2</font>)

= ab2 - a<font size="2">3</font> + b(0) + a3 - ab2

= 0

Therefore, (c - a) is a factor of given expression

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