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The polynomial f(x) is given by f(x) = x

4 + x
3 − 7x
2 + 3x + 2. If also,
f(x) = (x − 1)(x − 2)(x + 3)(x + c) + P x + Q, find the
values of P, Q and c

1 Answer

1 vote

To find the values of P, Q, and c, we need to equate the two expressions for f(x):

1. f(x) given as: f(x) = x^4 + x^3 - 7x^2 + 3x + 2

2. f(x) from the factored form: f(x) = (x - 1)(x - 2)(x + 3)(x + c) + Px + Q

Now, let's equate the two expressions:

x^4 + x^3 - 7x^2 + 3x + 2 = (x - 1)(x - 2)(x + 3)(x + c) + Px + Q

To find the values of P, Q, and c, we need to expand the right-hand side (RHS) of the equation and then match the coefficients of corresponding powers of x.

Expanding the right-hand side:

(x - 1)(x - 2)(x + 3)(x + c) = (x^2 - 2x - 3)(x^2 + cx - 2x - 3c) = x^4 + (c - 4)x^3 + (-2c - 4)x^2 + (-6 - 2c)x - 3c

Now, the equation becomes:

x^4 + x^3 - 7x^2 + 3x + 2 = x^4 + (c - 4)x^3 + (-2c - 4)x^2 + (-6 - 2c)x - 3c + Px + Q

Now, we can equate the coefficients of corresponding powers of x:

1. Coefficient of x^3:

1 = c - 4

c = 5

2. Coefficient of x^2:

-7 = -2c - 4

-7 = -2(5) - 4

-7 = -10 - 4

-7 = -14 (Not equal, this is incorrect)

Since the coefficient of x^2 does not match, there must be an error in the problem statement or the given polynomial.

Please double-check the given polynomial or the problem statement to ensure accuracy, or provide additional information to resolve any discrepancies.

User Hugos
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