To find the value of a and b, we can use polynomial division.
Given that x² + x - 6 is a factor of 2x⁴ + x³ - ax² + bx + a + b - 1, we can divide the polynomial 2x⁴ + x³ - ax² + bx + a + b - 1 by x² + x - 6 and set the remainder equal to zero. This will give us two equations that we can solve to find the values of a and b.
Performing polynomial division, we have:
2x² + 9x + a + b - 1
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x² + x - 6 | 2x⁴ + x³ - ax² + bx + a + b - 1
-(2x⁴ + 2x³ - 12x²)
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0 + 3x³ + (12 - a)x² + (b - 12)x + (a + b - 1)
Setting the remainder, 3x³ + (12 - a)x² + (b - 12)x + (a + b - 1), equal to zero, we get:
3x³ + (12 - a)x² + (b - 12)x + (a + b - 1) = 0
Now, comparing the coefficients of like terms on both sides of the equation, we can determine the values of a and b.
From the coefficient of x²:
12 - a = 0 -> a = 12
From the coefficient of x:
b - 12 = 0 -> b = 12
Therefore, the value of a is 12 and the value of b is also 12.
Answers & Comments
Answer:
To find the value of a and b, we can use polynomial division.
Given that x² + x - 6 is a factor of 2x⁴ + x³ - ax² + bx + a + b - 1, we can divide the polynomial 2x⁴ + x³ - ax² + bx + a + b - 1 by x² + x - 6 and set the remainder equal to zero. This will give us two equations that we can solve to find the values of a and b.
Performing polynomial division, we have:
2x² + 9x + a + b - 1
___________________________________
x² + x - 6 | 2x⁴ + x³ - ax² + bx + a + b - 1
-(2x⁴ + 2x³ - 12x²)
______________________
0 + 3x³ + (12 - a)x² + (b - 12)x + (a + b - 1)
Setting the remainder, 3x³ + (12 - a)x² + (b - 12)x + (a + b - 1), equal to zero, we get:
3x³ + (12 - a)x² + (b - 12)x + (a + b - 1) = 0
Now, comparing the coefficients of like terms on both sides of the equation, we can determine the values of a and b.
From the coefficient of x²:
12 - a = 0 -> a = 12
From the coefficient of x:
b - 12 = 0 -> b = 12
Therefore, the value of a is 12 and the value of b is also 12.