Using Factor Theorem, Show that a-b, b-c and c-a are factors of a(b²-c²) +b(c²-a²) + c(a²-b²)
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9th
Maths
Polynomials
Factorisation of Polynomial
Using factor theorem, show ...
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Asked on December 20, 2019 by
Barsha Sahu
Using factor theorem, show that a−b is a factor of a(b
2
−c
2
)+b(c
2
−a
2
)+c(a
2
−b
2
)
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ANSWER
We know that the factor theorem states that if the polynomial p(x) is divided by cx−d and the remainder, given by p(
c
d
), is equal to zero, then cx−d is a factor of p(x).
Consider the given expression a(b
2
−c
2
)+b(c
2
−a
2
)+c(a
2
−b
2
) and solve it as follows:
a(b
2
−c
2
)+b(c
2
−a
2
)+c(a
2
−b
2
)
=ab
2
−ac
2
+bc
2
−ba
2
+c(a−b)(a+b)(∵(x+y)(x−y)=x
2
−y
2
)
=ab
2
−ba
2
−ac
2
+bc
2
+c(a−b)(a+b)
=ab(b−a)−(a−b)c
2
+c(a−b)(a+b)
=−ab(a−b)−(a−b)c
2
+c(a−b)(a+b)
=(a−b)(−ab−c
2
+c(a+b))
=(a−b)(c(a+b)−ab−c
2
)
Hence, by factor theorem (a−b) is a factor of a(b
2
−c
2
)+b(c
2
−a
2
)+c(a
2
−b
2
).