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# 4. Factorise. (i) $$a^4 – b^4$$ (ii) $$p^4 – 81$$ (iii) $$x^4 – (y + z)^4$$ (iv) $$x^4 – (x – z)^4$$ (v) $$a^4 – 2a^2b^2 + b^4$$

(i) Given:$$a^4 – b^4$$

$$=(a^2)^2 – (b^2)^2\quad$$[∵ $$a^2 – b^2 = (a – b)(a + b)$$]

$$= (a^2 – b^2) (a^2 + b^2)$$

$$= (a – b) (a + b) (a^2 + b^2)$$

(ii) Given:$$p^4 – 81 = (p^2)^2 – (9)^2$$

$$= (p^2 – 9) (p^2 + 9)\quad$$[∵$$a^2 – b^2 = (a – b)(a + b)]$$

$$= (p – 3)(p + 3) (p^2 + 9)$$

(iii) Given:$$x^4 – (y + z)4$$ $$= (x^2)^2 – [(y + z)^2]^2\quad$$[∵ $$a^2 – b^2 = (a – b)(a + b)$$]

$$= [x^2 – (y + z)^2] [x^2 + (y + z)^2]$$

$$= [x – (y + z)] [x + (y + z)] [x^2 + (y + z)^2]$$

$$= (x – y – z) (x + y + z) [x^2 + (y + z)^2]$$

(iv) Given:$$x^4 – (x – z)^4 = (x^2)^2 – [(y – z)^2]^2$$

$$= [x^2 – (y – z)^2] [x^2 + (y – z)^2]$$

$$= (x – y + z) (x + y – z) (x^2 + (y – z)^2]$$

(v)Given: $$a^4 – 2a^2b^2 + b^4$$

$$= a^4 – a^2b^2 – a^2b^2 + b^4$$

$$= a^2(a^2 – b^2) – b^2(a^2 – b^2)$$

$$= (a^2 – b^2)(a^2 – b^2)$$

$$= (a^2 – b^2)^2$$

$$= [(a – b) (a + b)]^2$$

$$= (a – b)^2 (a + b)^2$$