Question

Factorise:
x⁴+x²y²+y²
x³-x²+ax+x-a-1
ab(x²+y²) + xy(a²+b²)
​

Answer 1

Step-by-step explanation:

[1] We have,

x⁴ + x²y² + y⁴

= (x⁴ + 2x²y² + y⁴) - x²y² {Adding and subtracting x²y²}

= (x² + y²)² - x²y²

= (x² + y²)² - (xy)²

= (x² + y² - xy) (x² + y² + xy) [Using a² - b² = (a + b) (a - b) to factor the expression]

Thus, x⁴ + x²y² + y⁴ = (x² + y² - xy) (x² + y² + xy)

[2] We have,

x³ - x² + ax + x - a - 1

= (x³ + ax + x) - x² - a - 1

= x(x² + a + 1) - 1(x² + a + 1)

= (x² + a + 1) (x - 1) [Factor out x² + a + 1 from the expression]

Thus, x³ - x² + ax + x - a - 1 = (x² + a +1) (x - 1)

[3] We have,

ab (x² + y²) + xy (a² + b²)

= abx² + aby² + a²xy + b²xy

= (abx² + a²xy) + (aby² + b²xy)

= ax(bx +ay) + by(ay + bx)

= (ay + bx) (ax + by) [Factor out ay + bx from the expression]

Thus, ab (x² + y²) + xy (a² + b²) = (ay + bx) (ax + by)