Given that the polynomial, x^3-3x^2-13x+15 x^3 -3x^2-13x+15, divides evenly for the

Question

divisor of x+3.
What do we know about the remainder?
(1 point)
𝑥
3 − 3𝑥 x − 3 − 13x + 15
2 − 13𝑥 + 15
3 x
2
the remainder is 1
the remainder is: x^2 − 6x + 5
x^2 − 6𝑥 + 5
We need more information to be able to determine the remainder.
the remainder is 0

Answer 1

Given that the polynomial \(x^3 - 3x^2 - 13x + 15\) divides evenly for the divisor \(x + 3\), we can apply the Remainder Theorem. The Remainder Theorem states that when a polynomial \(P(x)\) is divided by a linear divisor \(x - c\), the remainder is \(P(c)\).

In this case, we are dividing by \(x + 3\), which can be rewritten as \(x - (-3)\). Therefore, we need to evaluate \(P(-3)\):

\[ P(x) = x^3 - 3x^2 - 13x + 15 \]

Now, substituting \(-3\) into the polynomial:

\[ P(-3) = (-3)^3 - 3(-3)^2 - 13(-3) + 15 \] \[ = -27 - 3(9) + 39 + 15 \] \[ = -27 - 27 + 39 + 15 \] \[ = -54 + 54 \] \[ = 0 \]

Since \(P(-3) = 0\), this means that the remainder when dividing \(x^3 - 3x^2 - 13x + 15\) by \(x + 3\) is \(0\).

Therefore, the correct answer is:

the remainder is 0.