x^{4}-(x-z)^{4}

(x^{2})^{2}-{[(x-z)^{2}]^{2}}

or, x^{2}-(x-z)^{2}

or, [x-(x-z)][x+(x-z)]

or, (x-x+z)(x+x-z)

or, z(2x-z)

- -5

*x* ^{4} – (*x* – *z*)^{4}

= (*x* ^{2})^{2} – [(*x* – *z*)^{2}]^{2}

The given equation is of the form, *a* ^{2} – *b* ^{2} = (*a + b*) (*a – b*),

where *a* = *x* ^{2} and *b* = (*x* – *z*)^{2}

∴ (*x* ^{2})^{2} – [(*x* – *z*)^{2}]^{2 }= [*x* ^{2} + (*x* – *z*)^{2}] [*x* ^{2} – (*x* – *z*)^{2}]

= [*x* ^{2 }+ *x* ^{2} *+* *z* ^{2 }– 2*xz*] [*x* ^{2} – *x* ^{2} – *z* ^{2 }+ 2*xz*]

= [2*x* ^{2 }*+* *z* ^{2 }– 2*xz*] [2*x*– *z* ^{2}]

= [2*x* ^{2 }– 2*xz + *z^{2}] × *z*[2*x*– *z*]

Hence, *x* ^{4 }– (*x – z*)^{4 }= *z *(2*x* – *z *)(2*x* ^{2} – 2*xz* + *z* ^{2})

- 13

The expression

*x*^{4}– (*x*–*z*)^{4}can be factorised as,*x*

^{4}– (

*x*–

*z*)

^{4}

= (

*x*^{2})^{2}– {(*x*–*z*)^{2}}^{2}= [

*x*^{2}– (*x*–*z*)^{2}] [*x*^{2}+ (*x*–*z*)^{2}] [*a*^{2}–*b*^{2}= (*a*–*b*) (*a*+*b*)]= [

*x*– (*x*–*z*)] [*x*+ (*x*–*z*)] [*x*^{2}+*x*^{2}+*z*^{2}– 2*xz*][

*a*^{2}–*b*^{2}= (*a*–*b*) (*a*+*b*), (*a*–*b*)^{2}=*a*^{2}+*b*^{2}–2*ab*]= [

*x*–*x*+*z*)] [2*x*–*z*] [*x*^{2}+*x*^{2}+*z*^{2}– 2*xz*]=

*z*(2*x*–*z*) [2*x*^{2 }+*z*^{2}– 2*xz*]- 19

Ex: ax?+?by)2?+ (bx?+?ay)2

= [(ax)2?+ 2(ax)(by) + (by)2] + [(bx)2?+ 2(bx)(ay) + (ay)2]

= (a2x2?+ 2abxy?+?b2y2) + (b2x2?+ 2abxy?+?a2y2)ax?+?by)2?+ (bx?+?ay)2

= [(ax)2?+ 2(ax)(by) + (by)2] + [(bx)2?+ 2(bx)(ay) + (ay)2]

= (a2x2?+ 2abxy?+?b2y2) + (b2x2?+ 2abxy?+?a2y2)

=?a2x2?+?b2x2?+?a2y2?+?b2y2?+ 4abxy

=?x2(a2?+?b2) +?y2(a2?+?b2) + 4abxy

= (x2?+?y2) (a2?+?b2) + 4abxy

=?a2x2?+?b2x2?+?a2y2?+?b2y2?+ 4abxy

=?x2 un x value of 4(a2?+?b2) +?y2(a2?+?b3) + 4abxy

= (x2?+?y2) (a2?+?b2) + 4abxy

?

The factorise of x4 in every examples

:4abxy the factorise =is equal 4in every is called closure property

= [(ax)2?+ 2(ax)(by) + (by)2] + [(bx)2?+ 2(bx)(ay) + (ay)2]

= (a2x2?+ 2abxy?+?b2y2) + (b2x2?+ 2abxy?+?a2y2)ax?+?by)2?+ (bx?+?ay)2

= [(ax)2?+ 2(ax)(by) + (by)2] + [(bx)2?+ 2(bx)(ay) + (ay)2]

= (a2x2?+ 2abxy?+?b2y2) + (b2x2?+ 2abxy?+?a2y2)

=?a2x2?+?b2x2?+?a2y2?+?b2y2?+ 4abxy

=?x2(a2?+?b2) +?y2(a2?+?b2) + 4abxy

= (x2?+?y2) (a2?+?b2) + 4abxy

=?a2x2?+?b2x2?+?a2y2?+?b2y2?+ 4abxy

=?x2 un x value of 4(a2?+?b2) +?y2(a2?+?b3) + 4abxy

= (x2?+?y2) (a2?+?b2) + 4abxy

?

The factorise of x4 in every examples

:4abxy the factorise =is equal 4in every is called closure property

- 0