if the two factors of 2x^3+ax^2+bx+9 are x+3 and x-3 what are the values of a and b?
Answers
a = -1 & b = -18
Step-by-step explanation:
When x = -3
2(-3)³ + a(-3)² + b(-3) + 9 = 0
2(-27) + a(9) - 3b + 9 = 0
-54 + 9a - 3b + 9 = 0
9a - 3b = 45
3a - b = 15 --------(1)
Now , when x = 3
2(3)³ + a(3)² + b(3) + 9 = 0
2(27) + a(9) + 3b + 9 = 0
54 + 9a + 3b + 9 = 0
9a + 3b = -63
3a + b = -21 ----(2)
From equation (1) + equation (2)
6a = -6
a = -1
So , b = 3(-1) - 15
b = - 3 - 15
b = -18
Answer:
a = -1
b = -18
Note:
★ The possible values of the variable for which the polynomial becomes zero are called its zeros .
★ If x - a is a factor of the given polynomial p(x) , then x = a is a zero of p(x) and hence p(a) = 0.
Solution:
Here,
The given polynomial is ;
2x² + ax² + bx + 9 .
Also,
It is given that , x + 3 and x - 3 are the factors of the polynomial .
If x + 3 = 0 , then
x = -3
If x - 3 = 0 , then
x = 3
Thus,
At x = - 3 , 3 the polynomial becomes zero .
This,
=> 2(-3)³ + a(-3)² + b(-3) + 9 = 0
=> -54 + 9a - 3b + 9 = 0
=> -45 + 9a - 3b = 0
=> 3(-15 + 3a - b) = 0
=> -15 + 3a - b = 0
=> 3a - b = 15 -----------(1)
Also,
=> 2(3)³ + a(3)² + b(3) + 9 = 0
=> 54 + 9a + 3b + 9 = 0
=> 63 + 9a + 3b = 0
=> 3(21 + 3a + b) = 0
=> 21 + 3a + b = 0
=> 3a + b = -21 ------------(2)
Now,
Adding eq-(1) and (2) , we get ;
=> 3a - b + 3a + b = 15 + (-21)
=> 3a + 3a = 15 - 21
=> 6a = -6
=> a = -6/6
=> a = -1
Now,
Putting a = -1 in eq-(2) , we get ;
=> 3a + b = -21
=> 3(-1) + b = -21
=> -3 + b = -21
=> b = -21 + 3
=> b = -18