1. Factorize x^6 - 1
2a) Factorize x^4 + x^2 + 1 (Hints:Compare with x^4 + 2(x^2) + 1)
2b) Factorize (x+1)^4 + (x+1)^2 + 1 by using the result of (a).
Correct the mistake in question 3-4 of the following questions.
3. x^4 - 1
=(x^2)^2 - 1
=(x^2 + 1)(x^2 - 1)
4. x^3 - 27(y^3)
= (x - 3y)^3
Write down a polynomial of degree 4 with each of the following:
5. four different factors of degree 1
6. two different factors of degree 2
7. one factor of degree 2 and two factors of degree 1
F.2 / F.3 Maths about factorization (15X2=30)
2008-07-02 1:57 am
回答 (3)
2008-07-02 3:05 am
✔ 最佳答案
1. = x6 - 11. = (x3)2 - 1
1. = (x3 - 1)(x3 + 1)
1. = [(x - 1)(x2 + x + 1)][(x + 1)(x2 - x +1)]
1. = (x + 1)(x - 1)(x2 + x + 1)(x2 - x + 1)
2a. = X4 + x2 + 1
2a. = (x4 + 2x2 + 1) - x2
2a. = (x2 + 1)2 - x2
2a. = [(x2 + 1) + x][(x2 + 1) - x]
2a. = (x2 + x + 1)(x2 - x + 1)
2b. = (x+1)4 + (x+1)2 + 1
2b. = [(x+1)2 + (x+1) + 1][(x+1)2 - (x+1) + 1]
2b. = [x2 + 2x + 1 + x + 1 + 1][x2 + 2x + 1 - x - 1 + 1]
2b. = (x2 + 3x + 3)(x2 + x + 1)
3. = x4 - 1
3. = (x2)2 - 1)
3. = (x2 + 1)(x2 - 1)
3. = (x2 + 1)(x - 1)(x + 1)
4. = x3 - 27y3
4. = x3 - (3y)3
4. = (x - 3y)[x2 + x(3y) + (3y)3]
4. = (x - 3y)(x2 + 3xy + 9y2)
5. = x4 + 6x3 + 11x2 + 6x
5. = x(x+1)(x+2)(x+3)
6. = x4 + 3x2 + 2
6. = (x2 + 1)(x2 + 2)
7. = x4 + x3 + x2 + x
7. = x(x + 1)(x2 + 1)
=
2008-07-02 3:12 am
x6-1
=(x)-1
=(x-1)(x4+x+1)
=(x-1)(x+1)(x4+x+1)
x4+x+1
=x4+2x+1-x
=(x+1)-x
=(x+x+1)(x-x+1)
From (a), Let y=x+1
(x+1)4+(x+1)+1
=y4+y+1
=(y+y+1)(y-y+1)
=[(x+1)+(x+1)+1][(x+1)-(x+1)+1]
=(x+3x+3)(x+x+1)
3. The factorization process is not yet finished
4. x-27y=(x-3y)(x+3xy+9y) but not (x-3y)
5. a+b+c+d
6. a+b
7. a+b+c
=(x)-1
=(x-1)(x4+x+1)
=(x-1)(x+1)(x4+x+1)
x4+x+1
=x4+2x+1-x
=(x+1)-x
=(x+x+1)(x-x+1)
From (a), Let y=x+1
(x+1)4+(x+1)+1
=y4+y+1
=(y+y+1)(y-y+1)
=[(x+1)+(x+1)+1][(x+1)-(x+1)+1]
=(x+3x+3)(x+x+1)
3. The factorization process is not yet finished
4. x-27y=(x-3y)(x+3xy+9y) but not (x-3y)
5. a+b+c+d
6. a+b
7. a+b+c
2008-07-02 2:09 am
1. Factorize x^6 - 1
x6 - 1
=(x-1)(x2+x+1)
2a) Factorize x4 + x2 + 1 (Hints:Compare with x4 + 2(x2) + 1)
x4 + x2 + 1
=(x2+1)2
2b) Factorize (x+1)4 + (x+1)2 + 1 by using the result of (a).
(x+1)4 + (x+1)2 + 1
=[(x+1)2+1]2
Correct the mistake in question 3-4 of the following questions.
3. x4 - 1
=(x2)2 - 1
=(x2 + 1)(x2 - 1)
=(x2+1)(x-1)(x+1)
4. x3 - 27(y3)
=(x-3y)(x2+3xy+9y2)
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