Diketahui x₁, x₂ dan x₃ adalah akar-akar persamaan suku banyak x³ - 8x² + 9x + n = 0. Tentukan nilai n jika x₁ = 2x₂

<p>Jawaban : n = &nbsp;-846/343 dan n = 18.&nbsp;</p><p>&nbsp;</p><p>Ingat :</p><p>Jika x₁, x₂, x₃&nbsp;</p><p>adalah akar-akar persamaan polinomial ax³+bx²+cx+d = 0&nbsp;</p><p>maka :</p><p>x₁ + x₂ + x₃ = –b/a</p><p>x_{1 .}x₂ + x_{1 .}x_{3 &nbsp;}+ &nbsp;x_{2 .}x_3&nbsp; = &nbsp;c/a&nbsp;</p><p>x₁ . x₂ . x₃ = –d/a</p><p>&nbsp;</p><p>Pembahasan :</p><p>x³ - 8x² + 9x + n = 0. Tentukan nilai n jika x₁ = 2x₂</p><p>a = 1, b = -8, c = 9, d = n&nbsp;</p><p>maka:&nbsp;</p><p>x₁ + x₂ + x₃ = –b/a</p><p>2x₂ + x₂ + x₃ = –b/a</p><p>3x₂ + x₃ = 8</p><p>x₃ = 8 - 3x₂&nbsp;</p><p>&nbsp;</p><p>x_{1 .}x₂ + x_{1 .}x_{3 &nbsp;}+ &nbsp;x_{2 .}x_3&nbsp; = &nbsp;c/a&nbsp;</p><p>2x₂_.x₂ + 2x₂_.x_{3 &nbsp;}+ &nbsp;x_{2 .}x₃= c/a</p><p>2(x₂)²_&nbsp; + 3x₂_.x₃= 9</p><p>2(x₂)²_&nbsp; + 3x₂(8 - 3x₂)&nbsp;= 9</p><p>2(x₂)²_&nbsp; + 24x₂- 9(x₂)²&nbsp;= 9</p><p>-7(x₂)²_&nbsp; + 24x₂ - 9 = 0</p><p>-(7x₂²_&nbsp; - 24x₂ + 9) = 0</p><p>-(7x₂ - 3)(x₂ - 3) = 0</p><p>x₂ = 3/7 &nbsp;atau &nbsp;x₂ = 3</p><p>&nbsp;</p><p>Subtitusi ke &nbsp;x₁ = 2x₂ dan x₃ = 8 - 3x₂&nbsp;</p><p>Untuk x₂ = 3/7</p><p>x₁ = 2x₂ = 2.3/7 = 6/7</p><p>x₃ = 8 - 3x₂ = 8 - 3.3/7 = 8 - 8/7 = 47/7</p><p>&nbsp;</p><p>Untuk &nbsp;x₂ = 3</p><p>x₁ = 2x₂ = 2.3 = 6</p><p>x₃ = 8 - 3x₂ &nbsp;= 8 - 3.3 = 8 - 9 = -1</p><p>&nbsp;</p><p>Mencari nilai dari&nbsp;</p><p>&nbsp;x₁ . x₂ . x₃ = –d/a</p><ul><li>x₁ . x₂ . x₃ = –n</li></ul><p>6/7 . 3/7 . 47/7 = -n</p><p>n = - [ 6/7 . 3/7 . 47/7]</p><p>n = -846/343&nbsp;</p><p>&nbsp;</p><ul><li>x₁ . x₂ . x₃ = –n</li></ul><p>6 . 3 . -1 = -n</p><p>n = - (-18)</p><p>n = 18</p><p>&nbsp;</p><p>Sehingga nilai n yang memenuhi ada 2, yaitu ketika :</p><p>x₁ = 6/7, x₂ = 3/7, x₃ = 47/7 &nbsp;diperoleh n = -846/343 dan</p><p>x₁ = 6, x₂ = 3, x₃ = -1 &nbsp;diperoleh n = 18</p><p>_&nbsp;</p><p>Dengan demikian, nilai n adalah -846/343 dan 18.</p>