Solve the polynomial equation p(x)=x3+2x2−5x−6. Which of the following is a factor?(1 point)%0D%0AResponses%0D%0A%0D%0Ax−3%0D%0Ax minus 3%0D%0A%0D%0Ax−1%0D%0Ax minus 1%0D%0A%0D%0Ax+2%0D%0Ax plus 2%0D%0A%0D%0Ax+1

Answer 1

To determine which of the provided options is a factor of the polynomial \( p(x) = x^3 + 2x^2 - 5x - 6 \), we can use the Rational Root Theorem to test for possible rational roots. The factors of the constant term (-6) are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

We can evaluate the polynomial \( p(x) \) at these values:

1. Testing \( x = 1 \): \[ p(1) = 1^3 + 2(1^2) - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \quad (\text{not a root}) \]

2. Testing \( x = -1 \): \[ p(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \quad (\text{is a root}) \]

Since \( x = -1 \) is a root, then \( x + 1 \) is a factor of the polynomial.

Next, we can use polynomial long division or synthetic division to divide \( p(x) \) by \( x + 1 \) to find the other factors if needed, but for the question at hand, we can conclude that:

The answer is \( x + 1 \).