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Find the constants p and q such that when x³-px+q is divided by x²-3x+2, the remainder is 4x-1?

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more than 2 weeks ago

To find the constants p and q, we can use polynomial long division.

Divide x³ - px + q by x² - 3x + 2:

_______________________

x² - 3x + 2 | x³ + 0x² - px + q

- (x³ - 3x² + 2x)

_______________________

3x² - (p - 3)x + q

The remainder is 3x² - (p - 3)x + q.

We are given that the remainder is 4x - 1. Therefore, we can equate the corresponding coefficients:

3x² - (p - 3)x + q = 4x - 1

Comparing the coefficients of like terms, we get:

3 = 0 (coefficient of x²)

-(p - 3) = 4 (coefficient of x)

q = -1 (constant term)

From the first equation, we see that 3 = 0, which is not possible. Therefore, there are no constants p and q that satisfy the given conditions.

Divide x³ - px + q by x² - 3x + 2:

_______________________

x² - 3x + 2 | x³ + 0x² - px + q

- (x³ - 3x² + 2x)

_______________________

3x² - (p - 3)x + q

The remainder is 3x² - (p - 3)x + q.

We are given that the remainder is 4x - 1. Therefore, we can equate the corresponding coefficients:

3x² - (p - 3)x + q = 4x - 1

Comparing the coefficients of like terms, we get:

3 = 0 (coefficient of x²)

-(p - 3) = 4 (coefficient of x)

q = -1 (constant term)

From the first equation, we see that 3 = 0, which is not possible. Therefore, there are no constants p and q that satisfy the given conditions.

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