in a theatre, people are sitting in rows. If f(x) = x4+ 3x3+3x2+ ax +b represents the total number of people in the theatre and g(x) = x2+ 5 represents the number of people sitting in each row, then find number of rows in each class. Also find the value of a and b.
Answers
Given : people are sitting in rows.
f(x) = x⁴+ 3x³+3x²+ ax +b represents the total number of people in the theatre
g(x) = x²+ 5
To Find : number of rows
value of a and b.
Solution:
x² + 3x - 2
x²+ 5 _| x⁴+ 3x³+3x²+ ax +b |_
x⁴ + + 5x²
__________
3x³ - 2x²+ ax +b
3x³ + 15x +b
______________
- 2x²+ (a-15)x +b
-2x² -10
_______________
(a-15)x + b + 10
(a-15)x + b + 10 = 0
=> a - 15 = 0 => a = 15
b + 10 = 0 => b = -10
x² + 3x - 2 is number of Rows
value of a = 15
Value of b = - 10
Another method :
f(x) = x⁴+ 3x³+3x²+ ax +b
g(x) = x²+ 5
substitute x²+ 5 = 0 => x² = - 5 and Equate f(x) = 0
25 -15x -15 + ax + b = 0
=> (a - 15)x + (10 + b) = 0
=> a = 15 , b = -10
f(x) = x⁴+ 3x³+3x²+ 15x - 10
Then divided by = x²+ 5
x⁴+ 3x³+3x²+ 15x - 10 = (x²+ 5 )(x²+ 3x - 2)
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