If (x-2) and (x+3) is a factor of x3+ax2+bx-30, find a and b?
Answers
EXPLANATION.
(x - 2) & (x + 3) is a factor of the equation,
⇒ x³ + ax² + bx - 30.
As we know that,
(x - 2) is a factor of the polynomial,
⇒ x - 2 = 0.
⇒ x = 2.
put the value of x = 2 in equation, we get.
⇒ x³ + ax² + bx - 30.
⇒ (2)³ + a(2)² + b(2) - 30 = 0.
⇒ 8 + 4a + 2b - 30 = 0.
⇒ 4a + 2b - 22 = 0.
⇒ 2(2a + b - 11) = 0.
⇒ 2a + b - 11 0.
⇒ b = 11 - 2a. ⇒ (1).
(x + 3) is a factor of the polynomial
⇒ x + 3 = 0.
⇒ x = -3.
Put the value of x = -3 in equation, we get.
⇒ (-3)³ + a(-3)² + b(-3) - 30 = 0.
⇒ -27 + 9a - 3b - 30 = 0.
⇒ 9a - 3b - 57 = 0.
⇒ 3(3a - b - 19) = 0.
⇒ 3a - b - 19 = 0. ⇒ (2).
From equation (1) & (2) we get.
Put the value of equation (1) in equation (2), we get.
⇒ 3a - (11 - 2a) - 19 = 0.
⇒ 3a - 11 + 2a - 19 = 0.
⇒ 5a - 30 = 0.
⇒ 5a = 30.
⇒ a = 6.
Put the value of a = 6 in equation (1), we get.
⇒ b = 11 - 2a.
⇒ b = 11 - 2(6).
⇒ b = 11 - 12.
⇒ b = -1.
Value of a = 6 & b = -1.
Answer:
a = 6
b = -1
Step-by-step explanation:
Question:
If (x-2) and (x+3) is a factor of x^3+ax^2+bx-30, find a and b?
Solution:
(x-2)(x+3)
x-2=0 and x+3=0
x = 2 and x = -3
Now,
f(x) = x^3+ax^2+bx-30
f(2) = 2^3+2^2a+2b-30
f(2) = 8+4a+2b-30 = 0
4a + 2b = 22
2(2a+b) = 2(11)
2a+b = 11
Let it be first equation....
f(-3) = (-3)^3+(-3)^2a-3b-30 = 0
-27+9a-3b-30 = 0
9a-3b-57 = 0
9a-3b = 57
3a-b = 19
Let it be second equation:
Adding first and second equation:
2a+b+3a-b = 11+19
5a = 30
a = 6
So,
3a-b = 19
3(6) -b = 19
18 - b = 19
b = -1
Final answer:
a = 6
b = -1