Memasukkan x ke fungsi
(Soal OSP 2019)
f(x)= x⁴+4x³+6x²+4x+10
Maka,
[tex]f( \sqrt[4]{5} - 1)[/tex]
adalah?
A.
[tex]55 \sqrt[4]{5} + 86 \sqrt{2} - 3[/tex]
B. 90
C.
[tex]85 \sqrt{3} + 55 \sqrt[4]{55} - 65[/tex]
D.
[tex]85 \sqrt{3} + 5\sqrt[4]{55} + 65[/tex]
E. 14
E. 14
Penjelasan dengan langkah-langkah:
f(x)= x⁴+4x³+6x²+4x+10
x = [tex] \sqrt[4]{5}[/tex] -1
supaya lebih mudah, kita bagi-bagi dulu persamaan tersebut dalam beberapa bagian.
[tex] \: [/tex]
[tex]\begin{array}{l}{x}^{4} \\ = {( \sqrt[4]{5} - 1 )}^{4} \\ = {( {5}^{ \frac{1}{4} } - 1 )}^{4} \\ = ( {5}^{ \frac{1}{4} } - 1 ) ( {5}^{ \frac{1}{4} } - 1 )( {5}^{ \frac{1}{4} } - 1 )( {5}^{ \frac{1}{4} } - 1 )\\ = ( {5}^{ \frac{1}{4} } ) {}^{4} + {}_{4}C_{3}. ({5}^{ \frac{1}{4} } ) {}^{3} ( - 1) + {}_{4}C_{2}. ({5}^{ \frac{1}{4} } ) {}^{2} ( - 1) {}^{2} + {}_{4}C_{1}. ({5}^{ \frac{1}{4} } ) {}^{1} ( - 1) {}^{3} + ( - 1) {}^{4} \\ = 5 - 4.{5}^{ \frac{3}{4} } + 6. {5}^{ \frac{1}{2} } - 4. {5}^{ \frac{1}{4} } + 1 \\ = - 4 \sqrt[4]{125} + 6 \sqrt{5} - 4 \sqrt[4]{5}+6 \end{array}[/tex]
[tex] \: [/tex]
[tex]\begin{array}{l} 4x^{3} \\= 4( {5}^{ \frac{1}{4} } - 1 ) {}^{3} \\ = 4( {5}^{ \frac{1}{4} } - 1 ) ( {5}^{ \frac{1}{4} } - 1 )( {5}^{ \frac{1}{4} } - 1 )\\ = 4(( {5}^{ \frac{1}{4} } ) {}^{3} + {}_{3}C_{2}.( {5}^{ \frac{1}{4} } ) {}^{2} .( - 1) + {}_{3}C_{1}.( {5}^{ \frac{1}{4} } ) {}^{1} .( - 1) {}^{2} + ( - 1) {}^{3} ) \\ =4( {5}^{ \frac{3}{4} } - 3. {5}^{ \frac{1}{2} } + 3. {5}^{ \frac{1}{4} } - 1) \\ = 4 \sqrt[4]{125} - 12 \sqrt{5} + 12 \sqrt[4]{5} - 4\end{array}[/tex]
[tex] \: [/tex]
[tex]\begin{array}{l}6x^{2}\\=6( {5}^{ \frac{1}{4} } - 1) {}^{2} \\ = 6( ({5}^{ \frac{1}{4}} ) {}^{2} - 2. {5}^{ \frac{1}{4} } + 1) \\ = 6 \sqrt{5} - 12 \sqrt[4]{5} + 6 \end{array}[/tex]
[tex] \: [/tex]
[tex]\begin{array}{l}4x \\=4( \sqrt[4]{5} - 1) \\ = 4 \sqrt[4]{5} - 4\end{array}[/tex]
[tex] \: [/tex]
MAKA,
[tex]f( \sqrt[4]{5} - 1)[/tex]
adalah:
[tex] \begin{array}{l} f( \sqrt[4]{5} - 1) \\ = ( - 4 \sqrt[4]{125} + 6 \sqrt{5} - 4 \sqrt[4]{5}+6) + (4 \sqrt[4]{125} - 12 \sqrt{5} + 12 \sqrt[4]{5} - 4) + (6 \sqrt{5} - 12 \sqrt[4]{5} + 6) + (4 \sqrt[4]{5} - 4) + 10 \\ = 14 \end{array}[/tex]
Jawaban:
E. 14
Penjelasan dengan langkah-langkah:
mengubah bentuk persamaan pada pola bilangan segitiga pascal untuk memudahkan penghitungan terhadap nilai x yang diketahui
semoga sesuai harapan ^^
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