James says

Question

x+2 is a factor of x^3-3x+2
. Susan says he is incorrect and that x+2
is not a factor of x^3-3x+2
.

Determine which student is correct and explain how you know by using concepts learned in Unit 7. Also, show any work that is needed to support your explanation.

Answer 1

To determine whether \( x + 2 \) is a factor of \( x^3 - 3x + 2 \), we can use the Factor Theorem, which states that \( (x - c) \) is a factor of a polynomial \( P(x) \) if and only if \( P(c) = 0 \). In our case, we can rewrite \( x + 2 \) as \( x - (-2) \), indicating that we should check \( P(-2) \).

Let's evaluate \( P(-2) = (-2)^3 - 3(-2) + 2 \).

1. Calculate \( (-2)^3 \): \[ (-2)^3 = -8 \]

2. Calculate \( -3(-2) \): \[ -3(-2) = 6 \]

3. Now substitute these values back into \( P(-2) \): \[ P(-2) = -8 + 6 + 2 \]

4. Combine the terms: \[ P(-2) = -8 + 6 + 2 = -8 + 8 = 0 \]

Since \( P(-2) = 0 \), by the Factor Theorem, \( x + 2 \) is indeed a factor of \( x^3 - 3x + 2 \).

Conclusion

James is correct; \( x + 2 \) is a factor of \( x^3 - 3x + 2 \). We confirmed this by evaluating the polynomial at \( x = -2 \) and showing that it equals 0.