question_answer

Factorise:

(a) \[{{a}^{4}}-{{b}^{4}}\]

(b) \[{{p}^{4}}-81\]

(c) \[{{x}^{4}}-{{\left( y+z \right)}^{4~~}}\]

(d) \[{{x}^{4}}-{{(x-z)}^{4}}\]

Answer:

(a)    \[{{a}^{4}}-{{b}^{4}}={{({{a}^{2}})}^{2}}-{{({{b}^{2}})}^{2}}\]

\[=({{a}^{2}}-{{b}^{2}})({{a}^{2}}+{{b}^{2}})\]

\[=(a-b)(a+b)({{a}^{2}}+{{b}^{2}})\]

(b)            \[{{p}^{4}}-81={{({{p}^{2}})}^{2}}-{{(9)}^{2}}\]

\[=({{p}^{2}}-9)({{p}^{2}}+9)\]

\[=[{{(p)}^{2}}-{{(3)}^{2}}]({{p}^{2}}+9)\]

\[=(p-3)(p+3)({{p}^{2}}+9)\]

(c)              \[{{x}^{4}}-{{(y+z)}^{4}}={{({{x}^{2}})}^{2}}-{{\{{{(y+z)}^{2}}\}}^{2}}\]

\[=\{{{x}^{2}}-{{(y+z)}^{2}}\}\{{{x}^{2}}+{{(y+z)}^{2}}\}\]

\[=\{x-(y+z)\}\{x+(y+z)\}\{{{x}^{2}}+{{(y+z)}^{2}}\}\]

\[=(x-y-z)(x+y+z)\{{{x}^{2}}+{{(y+z)}^{2}}\}\]

(d)  \[{{x}^{4}}-{{(x-z)}^{4}}={{({{x}^{2}})}^{2}}-{{\{{{(x-z)}^{2}}]}^{2}}\]

\[=\{{{x}^{2}}-{{(x-z)}^{2}}\}\{{{x}^{2}}+{{(x-z)}^{2}}\}\]

\[=(x-x+z)(x+x-z)\{{{x}^{2}}+{{(x-z)}^{2}}\}\]