Question

Resolve into factors: b(a+c) + c(a+b)+b^2+c^2.

Answer 1

Answer:

$a^2(b - c) + b^2(c-a) + c^2(a - b) \\ = a^2b - a^2c + b^2c - b^2a +c^2a-c^2b \\ =a^2b - b^2a - a^2c + b^2c + c^2a -c^2b \\ = ab(a - b) - c(a^2 - b^2) +c^2(a - b) \\ =ab(a - b) -c(a - b)(a + b) +c^2(a - b) \\ =(a - b)(ab - c(a + b) +c^2) \\ =(a - b)(ab - cb -ca + c^2) \\ = (a - b)(b(a-c) -c(a-c)) \\ =(a-b)(b-c)(a-c)$

Answer 2

Answer:

Answer:

\begin{gathered}a^2(b - c) + b^2(c-a) + c^2(a - b) \\ = a^2b - a^2c + b^2c - b^2a +c^2a-c^2b \\ =a^2b - b^2a - a^2c + b^2c + c^2a -c^2b \\ = ab(a - b) - c(a^2 - b^2) +c^2(a - b) \\ =ab(a - b) -c(a - b)(a + b) +c^2(a - b) \\ =(a - b)(ab - c(a + b) +c^2) \\ =(a - b)(ab - cb -ca + c^2) \\ = (a - b)(b(a-c) -c(a-c)) \\ =(a-b)(b-c)(a-c)\end{gathered}

a

2

(b−c)+b

2

(c−a)+c

2

(a−b)

=a

2

b−a

2

c+b

2

c−b

2

a+c

2

a−c

2

b

=a

2

b−b

2

a−a

2

c+b

2

c+c

2

a−c

2

b

=ab(a−b)−c(a

2

−b

2

)+c

2

(a−b)

=ab(a−b)−c(a−b)(a+b)+c

2

(a−b)

=(a−b)(ab−c(a+b)+c

2

)

=(a−b)(ab−cb−ca+c

2

)

=(a−b)(b(a−c)−c(a−c))

=(a−b)(b−c)(a−c)