Using the remainder theorum determine the remainder when xcubed PLUS 3xsquared - x - 2 is divided by x PLUS 3 TIMES x PLUS 5 theres brackets around x PLUS 3 and seperatly around x PLUS 5?

Answer 1

(x^3 + 3x^2 - x - 2)/[(x + 3)(x + 5) in this case you can use the long division to divide polynomials and to find the remainder of this division. But you cannot use neither the synthetic division to divide polynomials nor the Remainder theorem to determine the remainder. You can use both the synthetic division and the Remainder theorem only if the divisor is in the form x - c. In this case the remainder must be a constant because its degree is less than 1, the degree of x - c.

The remainder theorem says that if a polynomial f(x) is divided by x - c, then the remainder is f(c).

If the question is to determine the remainder by using the remainder theorem, then you are asking to find the value of f(-3) when you are dividing by x + 3, or f(-5)when you are dividing by x + 5 . Just substitute -3 or -5 with x into the dividend x^3 + 3x^2 - x - 2, and you can find directly the value of the remainder.

f(-3) = (-3)^3 + 3(-3)^2 - (-3) - 2 = -27 + 27 + 3 - 2 = 1 (remainder is 1)

f(-5) = (-5)^3 + 3(-5)^2 - (-5) - 2 = -125 + 75 + 5 - 2 = -47 (remainder is -47).