Kesimpulan :
i) 3 akar real positif dan 1 akar real negatif.
ii) 1 akar real positif, 1 akar real negatif, dan 2 akar kompleks sekawan.
Latihan Soal !
1. Selesaikan persamaan :
x+ex = 0 dengan range x = [ 1,0] !
2. Selesaikan persamaan
xe-x +1 = 0!
3. Lokalisasikan akar-akar polinom = 3+ 2 1
Dari tabulasi terlihat bahwa terjadi pergantian tanda dua kali yaitu pada selang ( 2, 1) dan (1,2). Kesimpulan komposisi akar adalah:
1 akar real negatif
1 akar real positif
2 akar kompleks sekawan
Jadi kemungkinan komposisi diatas yang benar adalah ii).
( )
2
17
1
1
0
1
1
1
2
5
Selang akar
Agar selang akar dapat ditentukan, harus dihitung r sebagai berikut:
=1+ 11. 11. 11.11
=1+1=2
Jadi akarnya terletak dalam selang 2,2
b. akar real negatif
p (-x) = 4 + 3 - 2+ – 1, sehingga pola tanda koefisiennya adalah
+ + , jadi =1
Kita peroleh: =0. Akibatnya mempunyai nilai 1
Persamaan kuadrat = 2+ + =0 dengan 0, rumus akarnya adalah :
12= ± 2 4 2
Bentuk persamaan :
Persamaan aljabar
Persamaan transenden
Persamaan campuran
1. Pengertian Persamaan dan Akarnya
2. Lokalisasi Akar
Cara untuk melokalisasikan akar antara lain :
lokalisasi akar secara grafik
lokalisasi akar secara tabulasi
lokalisasi akar untuk persamaan polinom
2.1 Lokalisasi Akar Secara Grafik
Akar tidak dapat muncul dalam suatu selang yang ditentukan oleh batas bawah 1 dan batas atas 2
0
Y
X
Melukiskan kasus dimana satu akar tunggal dikurung oleh nilai – nilai ( ) yang positif dan negatif
0
X
Y
( 1) dan ( 2) berseberangan terhadap sumbu , memperlihatkan tiga akar muncul pada selang itu.
Jika ( ) dan ( ) bertanda sama, maka diantara nilai – nilai tersebut tidak terdapat akar atau terdapat akar yang banyaknya genap
0
X
Y
0
X
Y
Contoh :
Lokalisasikan selang akar yang mengandung akar dari = dengan cara grafik tunggal dan grafik ganda !
Penyelesaian :
Cara grafik tunggal
Karena grafiknya memotong sumbu antara =0 dan =1 , berarti terdapat satu akar pada selang (0,1)
0
1
2
1
akar
b. Cara grafik ganda
Terlihat bahwa absis titik potong kedua grafik diatas berada pada selang ( 0,1 )
0
1
2
1
2.2 Lokalisasi Akar Secara Tabulasi
Nilai – nilai fungsi pada selang yang ditentukan dihitung memakai suatu lebar selang tertentu. Untuk memudahkan penulisannya, nilai – nilai tersebut dituliskan dalam suatu bentuk tabulasi
contoh :
Tentukan selang yang mengandung salah satu akar fungsi =
AKAR PERSAMAAN TAK LINEAR
MATEMATIKA 3A
Prila Ari Sandy (12511001)
Dera Arvina P (12511009)
Risa Rodianti (12511018)
Euis Lela N (12511021)
Devianti Rochman (12511023)
Penyelesaian :
Karena untuk =0,4, nilai >0 dan untuk =0,6, nilai <0 maka terdapat salah satu akarnya pada selang 0,4;0,6
0,0
1,00
0,2
0,619
0,4
0,270
0,6
0,251
0,8
0,351
Contoh : Lokasikan akar-akar polinom
= 4 3 2+ 1
Penyelesaian :
Aturan tanda descartes
a. akar real positif
Pola tanda koefisien
: + + . Jadi = 3.
Kita peroleh: =0 atau =2. Akibatnya mempunyai peluang dua nilai yaitu =3 atau =1
Lokalisasi Akar untuk Persamaan Polinom
Bentuk baku persamaan polinom berderajat n:
= 0+ 1 + 2 2+…+
dengan 0
Untuk melokasikan akar-akarnya yang real digunakan beberapa sifat akar, yaitu:
a) Aturan tanda Descartes
Akar real positif, berlaku:
= 0, 2, 4, …
Akar real negatif, berlaku:
= 0,2,4,…
b) Selang Akar
Andaikan = 1+ 1
Dengan adalah koefisien ( ) dan koefisien pangkat tinggi dari ( ), maka semua akar ( ) akan terletak pada selang [ , ]
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