Question

Factorise : (a² − b²) c + (b² − c²)a 

Answer 1

(a² − b²) c + (b² − c²)a

Step 1: Write down the expression

(a² − b²) c + (b² − c²)a

Step 2: Use the difference of squares formula

x² − y² = (x − y)(x + y)

So,

a² − b² = (a − b)(a + b)

and 

b² − c² = (b − c)(b + c)

Substitute these into the given expression:

= (a − b)(a + b)c + (b − c)(b + c)a

Step 3: Expand both terms

= c(a² − b²) + a(b² − c²)

Simplify by removing brackets:

= a²c − b²c + ab² − ac²

Step 4: Group terms for common factors

Group as:

= (a²c − ac²) + (ab² − b²c)

Step 5: Factor each group

From the first group, factor out a c:

ac(a − c)

From the second group, factor out b²:

b2(a − c)

Now the expression becomes: 

= ac(a − c) + b²(a − c)

Step 6: Factor out the common term (a−c)

= (a − c)(ac + b²)