Anonymous
Anonymous asked in 科學數學 · 1 decade ago

Mathemaics

1. (a) Factorize the following expressions.

(i) x^4-1 [Hint: x^4=(x^2)^2]

(ii) X^8-1 [Hint: x^8=(x^4)^2]

(iii) (x^2+1)^2-4x^2

(b) Simplify (x^8-1)/[(x^2+1)^2-4x^2]×(x^4-1)/(x^4+1).

2 Answers

Rating
  • 1 decade ago
    Favorite Answer

    ( a^2 - b^2 ) = ( a + b) ( a - b)

    i. x^4 - 1 = [ ( x^2 )^2 - 1 ] = ( x^2 + 1 ) ( x^2 - 1 )

    = ( x^2 + 1 ) ( x + 1 ) ( x - 1)

    ii. x^8 - 1 = ( x^4 + 1 ) ( x^4 - 1 ) = ( x^4+ 1)( x^2 + 1 ) ( x + 1 ) ( x - 1)

    iii ( x^2 + 1 )^2 - 4x^2

    = ( x^2 + 1 )^2 - ( 2x )^2 = ( x^2 + 2x + 1 ) ( x^2 - 2x + 1 )

    = ( x + 1 )^2 * ( x - 1 )^2 = ( x^2 - 1 )^2

    b . 將題目分類依下

    => ( x^8 - 1 ) / ( x^4 + 1 ) * ( x^4 - 1 ) / [(x^2+1)^2-4x^2]

    = ( x^4 - 1 ) * ( x^4 - 1 ) / ( x^2 - 1 )^2

    = ( x^2 + 1 )^2 * ( x^2 - 1 )^2 / ( x^2 - 1 )^2

    = ( x^2 + 1 )^2

    Source(s): myself
  • ?
    Lv 7
    1 decade ago

    (i) x^4-1

    =(x^2-1)(x^2+1)

    =(x-1)(x+1) (x^2+1)

    Ans: (x-1)(x+1) (x^2+1)

    (ii) X^8-1=(x^4)^2-1

    =(x^4-1) (x^4+1)

    =(x^2-1)(x^2+1) (x^4+1)

    =(x-1)(x+1)(x^2+1) (x^4+1)

    Ans: (x-1)(x+1)(x^2+1) (x^4+1)

    (iii) (x^2+1)^2-4x^2 ………..A^2-B^2=(A-B)(A+B)

    =(X^2+1-2X)( X^2+1+2X)

    =(X-1)^2(X+1)^2

    Ans: (X-1)^2(X+1)^2

    Simplify (x^8-1)/[(x^2+1)^2-4x^2](x^4-1)/(x^4+1)

    =(x^4-1) (x^4+1)/[(x^2+1-2X)(X^2+1+2X)](x^4-1)/(x^4+1)

    =(x^4-1) (x^4+1)/[(x-1)^2(X+1)^2](x^4-1)/(x^4+1)

    =(x^2-1)(X^2+1) (x^4+1) (x^4-1) / (x-1)^2(X+1)^2(x^4+1)

    =(x-1)(x+1) (X^2+1) (x^4+1) (x-1)(x+1)(x^2+1)/ (x-1)^2(X+1)^2(x^4+1)

    =(x^2+1)^2

    Ans: (x^2+1)^2

    Source(s): <自已>
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