Question

Suku banyak p(x)=x3+ax2+bx-2 bersisa 7 dibagi 2x-3 dan bersisa 0 jika dibagi x+2. Maka nilai a+b=.... Mohon bantuannya dalam mengerjakan soal ini...(Soal UM Undip 2015 kode 543)

Answer 1

Verified answer

Bab Polinomial
Matematika SMA Kelas XI

2x - 3 = 0
x = 3/2

x + 2 = 0
x = -2

p(x) = x³ + ax² + bx - 2 = 7
p(3/2) = (3/2)³ + a(3/2)² + b . (3/2) - 2 = 7
27/8 + 9a/4 + 3b/2 = 7 + 2
***dikali 8***
27 + 18a + 12b = 72
18a + 12b = 72 - 27
18a + 12b = 45
***dibagi 3***
6a + 4b = 15

p(-2) = (-2)³ + a . (-2)² + b (-2) - 2 = 0
-8 + 4a - 2b = 2
4a - 2b = 2 + 8
***dikali 2***
8a - 4b = 20

8a - 4b = 20
6a + 4b = 15
-------------------- +
       14a = 35
           a = 35/14
           a = 5/2

6a + 4b = 15
6 . (5/2) + 4b = 15
4b = 15 - 15
b = 0/4
b = 0

a + b = 5/2 + 0 = 5/2

Answer 2

Verified answer

Untuk x = 3/2
p(3/2) = (3/2)³ + a(3/2)² + b(3/2) - 2
7 = (27/8) + a(9/4) + 3b/2 - 2
9a/4 + 3b/2 = 7 + 2 - 27/8
9a/4 + 3b/2 = 72/8 - 27/8
½(9a/2 + 3b) = 45/8
9a/2 + 3b = 45/4
3(3a/2 + b) = 45/4
3a/2 + b = 15/4

untuk x = -2
p(x) = (-2)³ + a(-2)² + b(-2) - 2
0 = -8 + 4a - 2b - 2
4a - 2b = 8 + 2
4a - 2b = 10
2(2a - b) = 10
2a - b = 5

metode eliminasi
3a/2 + b = 15/4
2a - b = 5
____________+
7a/2 = 35/4
a = 35/4 × 2/7
a = 70/28
a = 5/2

metode subtitusi
2a - b = 5
b = 2a - 5
b = 2(5/2) - 5
b = 5 - 5
b = 0

maka nilai dari
= a + b
= 5/2 + 0
= 5/2
= 2,5

semoga berguna +_+