F.3 MATH (因式分解)

應用公式：
u² - v² = (u + v)(u - v)
u2 + 2uv + v2 = (u + v)2
u2 - 2uv + v2 = (u - v)2


1.
設 u = 4 及 v = k²
48 - 3k^4
= 3(16 - k^4)
= 3[(4)² - (k²)²]
= 3(u² - v²)
= 3(u + v)(u - v)
= 3(4 + k²)(4 - k²)
= 3(4 + k²)[(2)² - k²]
= 3(4 + k²)(2 + k)(2 - k) ...... (答案)
（題目所給的答案錯誤。）


2.
設 u = 8mn 及 v = m + n
64m²n² - (m + n)²
= (8mn)² - (m + n)²
= u² - v²
= (u + v)(u - v)
= [8mn + (m + n)] [8mn - (m + n)]
= (8mn + m + n)(8mn - m - n) ...... (答案)


3.
設 u = 2x 及 v = 3y
12x² - 27y²
= 3(4x² - 9y²)
= 3[(2x)² - (3y)²]
= 3(u² - v²)
= 3(u + v)(u - v)
= 3(2x + 3y)(2x - 3y) ...... (答案)


4.
x² + 2x + 1
= (x)² + 2(x)(1) + (1)²
= (x + 1)² ...... (答案)


5.
49 + 14m + m²
= (7)² + 2(7)(m) + (m)2
= (7 + m)² ..... (答案)


6.
4x² + 20x + 25
= (2x)² + 2(2x)(5) + (5)²
= (2x + 5)² ...... (答案)


7.
8a² - 56a + 98
= 2(4a² - 28a + 49)
= 2[(2a)² - 2(2a)(7) + (7)²]
= 2(2a - 7)² ...... (答案)


8.
75m² - 30mn + 3n²
= 3(25m² - 10mn + n²)
= 3[(5m)² - 2(5m)(n) + (n)²]
= 3(5m - n)² ...... (答案)


9.
16a² - 8abc + b²c²
= (4a)² - 2(4a)(bc) + (bc)²
= (4a - bc)² ...... (答案)

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Hope I can help you !


9. 16a^2 - 8abc + b^2 c^2

= (4a)^2 - 2(4a)(bc) + (bc)^2 * a^2 - 2ab + b^2

= (4a - bc)^2 = (a-b)^2



8. 75m^2 - 30mn + 3n^2

= 3 ( 25m^2 - 10mn + n^2 )

= 3 ((5m)^2- 2(5m)(n) + n^2)


7. 8a^2 - 56a + 98

= 2 ( 4a^2 - 28a + 49 )

= 2(2a - 7)^2


6. 4x^2 + 20x + 25

= (2x)^2 + 2(2x)(5) + 5^2 * a^2 + 2ab+ b^2

= (2x + 5)^2 = (a+b)^2



4. x^2 + 2x + 1

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

= (x + 1)^2



5. 49 + 14m + m^2

= m^2 + 14m + 49

= m^2 + 2(m)(7) + 7^2

= (m + 7)^2