4. Factorise.
(i) \(a^4 – b^4\)

(ii) \(p^4 – 81\)

(iii) \(x^4 – (y + z)^4\)

(iv) \(x^4 – (x – z)^4\)

(v) \(a^4 – 2a^2b^2 + b^4\)


Answer :

(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: \(a^4 – 2a^2b^2 + b^4\)

\(= a^4 – a^2b^2 – a^2b^2 + b^4\)

\(= a^2(a^2 – b^2) – b^2(a^2 – b^2)\)

\(= (a^2 – b^2)(a^2 – b^2)\)

\(= (a^2 – b^2)^2\)

\(= [(a – b) (a + b)]^2\)

\(= (a – b)^2 (a + b)^2\)

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