The polynomial x^4 + 6x^3 + 8x^2 - ax + b is divisible by (x + 2). If the polynomial is divided by (x - 2), the remainder will be 84. Find the values of a and b.
Answer (2)
The values of a and b in the polynomial are a = 3 and b = − 6 . These were determined using the Remainder Theorem by evaluating the polynomial at specific values. The conditions for divisibility and the remainder allowed us to set up a system of equations for solving these variables.
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To solve this problem, we need to find the values of a and b in the polynomial f ( x ) = x 4 + 6 x 3 + 8 x 2 − a x + b given two conditions.
Condition 1: The polynomial is divisible by ( x + 2 ) .
Since f ( x ) is divisible by ( x + 2 ) , by the Factor Theorem, f ( − 2 ) = 0 . Let's substitute x = − 2 into the polynomial and set it equal to zero:
f ( − 2 ) = ( − 2 ) 4 + 6 ( − 2 ) 3 + 8 ( − 2 ) 2 − a ( − 2 ) + b = 0
Simplifying:
16 − 48 + 32 + 2 a + b = 0
2 a + b = 0 ... (Equation 1)
Condition 2: The remainder is 84 when divided by ( x − 2 ) .
By the Remainder Theorem, f ( 2 ) = 84 . Substitute x = 2 into the polynomial:
f ( 2 ) = ( 2 ) 4 + 6 ( 2 ) 3 + 8 ( 2 ) 2 − a ( 2 ) + b = 84
Simplifying:
16 + 48 + 32 − 2 a + b = 84
96 − 2 a + b = 84
− 2 a + b = − 12 ... (Equation 2)
Next, we solve the system of equations formed by Equations 1 and 2:
Equation 1: 2 a + b = 0
Equation 2: − 2 a + b = − 12
Subtract Equation 2 from Equation 1:
( 2 a + b ) − ( − 2 a + b ) = 0 + 12
This simplifies to:
4 a = 12
Solving for a :
a = 4 12 = 3
Substitute a = 3 back into Equation 1:
2 ( 3 ) + b = 0
6 + b = 0
b = − 6
Therefore, the values of a and b are 3 and -6, respectively.
