1 a) Expand the following.
(i) (x + y)(x^ - xy + y^)
(ii) (x - y)(x^ + xy - y^)
b) Using the result fo (a), expand (z + 1)(z - 1)(z^ +z + 1)(z^ - z + 1).
c) Using the result fo (b), express the following as the product of prime factors.
(i) 3 6次方 - 1
(ii) 4 6次方 - 1
急急急...f.maths
2007-10-06 7:12 pm
回答 (3)
2007-10-06 7:19 pm
✔ 最佳答案
(i) (x + y)(x^ - xy + y^) = x^3 + y^3(ii) (x - y)(x^ + xy - y^) = x^3 - y^3 (Direct expansion only)
b)
(z + 1)(z - 1)(z^ +z + 1)(z^ - z + 1)
= [(z + 1)(z^ - z + 1)] [(z - 1)(z^ +z + 1)]
= (z^3 - 1)(z^3 + 1)
= z^6 - 1
c)
(i) 3^6 - 1
Put z = 3
3^6 - 1 = (3 + 1)(3 - 1)(9 + 3 + 1)(9 - 3 + 1)
= 2*4*13*7
= 2^3 * 13*7
(ii) Put z = 4
4^6 - 1 = (4 + 1)(4 - 1)(16 + 4 + 1)(16 - 2 + 1)
=3*5*(21)*(13)
= 3^2 *5 * 7 *13
2007-10-06 7:49 pm
i) (x + y)(x^ - xy + y^) = x^3 + y^3
(ii) (x - y)(x^ + xy - y^) = x^3 - y^3 (Direct expansion only)
b)
(z + 1)(z - 1)(z^ +z + 1)(z^ - z + 1)
= [(z + 1)(z^ - z + 1)] [(z - 1)(z^ +z + 1)]
= (z^3 - 1)(z^3 + 1)
= z^6 - 1
c)
(i) 3^6 - 1
Put z = 3
3^6 - 1 = (3 + 1)(3 - 1)(9 + 3 + 1)(9 - 3 + 1)
= 2*4*13*7
= 2^3 * 13*7
(ii) Put z = 4
4^6 - 1 = (4 + 1)(4 - 1)(16 + 4 + 1)(16 - 2 + 1)
=3*5*(21)*(13)
= 3^2 *5 * 7 *13
(ii) (x - y)(x^ + xy - y^) = x^3 - y^3 (Direct expansion only)
b)
(z + 1)(z - 1)(z^ +z + 1)(z^ - z + 1)
= [(z + 1)(z^ - z + 1)] [(z - 1)(z^ +z + 1)]
= (z^3 - 1)(z^3 + 1)
= z^6 - 1
c)
(i) 3^6 - 1
Put z = 3
3^6 - 1 = (3 + 1)(3 - 1)(9 + 3 + 1)(9 - 3 + 1)
= 2*4*13*7
= 2^3 * 13*7
(ii) Put z = 4
4^6 - 1 = (4 + 1)(4 - 1)(16 + 4 + 1)(16 - 2 + 1)
=3*5*(21)*(13)
= 3^2 *5 * 7 *13
2007-10-06 7:24 pm
(x + y)(x2 - xy + y2)
=x3-x2y+xy2+x2y-xy2+y3
=x3 +y3
(x - y)(x2 + xy - y2)
=x3+x2y-xy2-x2y+xy2-y3
=x3 -y3
(z +1) (z -1)(z2 -z +1) (z2 +z -1)
From the above, sub x=z, y=1
(z +1) (z -1)(z2 -z +1) (z2 +z -1)
= (z3 +13) (z3 -1)
=z6-1
Sub z=3
36-1
=(3 +1) (3 -1)(32 -3 +1) (32 +3 -1)
=(4)(2)(7)(11)
Sub z=4
46-1
=(4 +1) (4 -1)(42 -4 +1) (42 +4 -1)
=(5)(3)(13)(19)
=x3-x2y+xy2+x2y-xy2+y3
=x3 +y3
(x - y)(x2 + xy - y2)
=x3+x2y-xy2-x2y+xy2-y3
=x3 -y3
(z +1) (z -1)(z2 -z +1) (z2 +z -1)
From the above, sub x=z, y=1
(z +1) (z -1)(z2 -z +1) (z2 +z -1)
= (z3 +13) (z3 -1)
=z6-1
Sub z=3
36-1
=(3 +1) (3 -1)(32 -3 +1) (32 +3 -1)
=(4)(2)(7)(11)
Sub z=4
46-1
=(4 +1) (4 -1)(42 -4 +1) (42 +4 -1)
=(5)(3)(13)(19)
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