We express both terms as perfect squares: x^4 becomes (x²)² and (x - z)^4 becomes ((x - z)²)². This allows us to apply the algebraic identity a² - b² = (a - b)(a + b) with a = x² and b = (x - z)².

We express both terms as perfect squares: x^4 becomes (x²)² and (x - z)^4 becomes ((x - z)²)². This allows us to apply the algebraic identity a² - b² = (a - b)(a + b) with a = x² and b = (x - z)².

Applying the difference of squares identity a² - b² = (a - b)(a + b), the expression factors into the product of [x² - (x - z)²] and [x² + (x - z)²].

Now we expand the second factor using (x - z)² = x² - 2xz + z². Adding x² gives x² + x² - 2xz + z² = 2x² - 2xz + z².

x raise to power 4 -(x-z) 4 . Fazctorise

Answer: $z(2x - z)(2x^2 - 2xz + z^2)$

Step-by-step solution

Step 1: Rewrite as difference of squares

We express both terms as perfect squares: $x^4$ becomes $(x^2)^2$ and $(x - z)^4$ becomes $((x - z)^2)^2$ . This allows us to apply the algebraic identity $a^2 - b^2 = (a - b)(a + b)$ with $a = x^2$ and $b = (x - z)^2$ .

Step 2: Apply identity $a^2 - b^2$

Applying the difference of squares identity $a^2 - b^2 = (a - b)(a + b)$ , the expression factors into the product of $[x^2 - (x - z)^2]$ and $[x^2 + (x - z)^2]$ .

Step 3: Factor the first factor using difference of squares

The first factor $x^2 - (x - z)^2$ is also a difference of two squares. Using the same identity again with $a = x$ and $b = x - z$ , we get $[x - (x - z)][x + (x - z)] = (x - x + z)(x + x - z) = z(2x - z)$ .

Step 4: Expand and simplify the second factor

Now we expand the second factor using $(x - z)^2 = x^2 - 2xz + z^2$ . Adding $x^2$ gives $x^2 + x^2 - 2xz + z^2 = 2x^2 - 2xz + z^2$ .

Step 5: Combine all factors

Combining the results of both factors, the complete factorisation of $x^4 - (x - z)^4$ is $z(2x - z)(2x^2 - 2xz + z^2)$ .

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