A factor of the expression $x^3 – 2x + 1$ is :

Q: A factor of the expression $x^3 – 2x + 1$ is :

To determine why Option 1, $x - 1$, is a factor of the expression $x^3 - 2x + 1$, we can use the Factor Theorem. The Factor Theorem states that if $x - c$ is a factor of a polynomial $f(x)$, then $f(c) = 0$. 1. **Verification of Option 1: $x - 1$** - Substitute $x = 1$ into the polynomial $f(x) = x^3 - 2x + 1$: \[ f(1) = 1^3 - 2(1) + 1 = 1 - 2 + 1 = 0 \] - Since $f(1) = 0$, by the Factor Theorem, $x - 1$ is indeed a factor of the polynomial $x^3 - 2x + 1$. 2. **Explanation for why other options are incorrect:** - **Option 2: $x + 1$** - Substitute $x = -1$ into the polynomial: \[ f(-1) = (-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2 \] - Since $f(-1) \neq 0$, $x + 1$ is not a factor. - **Option 3: $x + 2$** - Substitute $x = -2$ into the polynomial: \[ f(-2) = (-2)^3 - 2(-2) + 1 = -8 + 4 + 1 = -3 \] - Since $f(-2) \neq 0$, $x + 2$ is not a factor. - **Option 4: $x - 2$** - Substitute $x = 2$ into the polynomial: \[ f(2) = 2^3 - 2(2) + 1 = 8 - 4 + 1 = 5 \] - Since $f(2) \neq 0$, $x - 2$ is not a factor. In conclusion, the only option that satisfies the condition of the Factor Theorem is Option 1, $x - 1$. Therefore, $x - 1$ is the correct factor of the expression $x^3 - 2x + 1$.

Question

Accepted Answer

To determine why Option 1, $x - 1$, is a factor of the expression $x^3 - 2x + 1$, we can use the Factor Theorem. The Factor Theorem states that if $x - c$ is a factor of a polynomial $f(x)$, then $f(c) = 0$.

1. **Verification of Option 1: $x - 1$**
   - Substitute $x = 1$ into the polynomial $f(x) = x^3 - 2x + 1$:
     $$
     f(1) = 1^3 - 2(1) + 1 = 1 - 2 + 1 = 0
     $$
   - Since $f(1) = 0$, by the Factor Theorem, $x - 1$ is indeed a factor of the polynomial $x^3 - 2x + 1$.

2. **Explanation for why other options are incorrect:**

- **Option 2: $x + 1$**
     - Substitute $x = -1$ into the polynomial:
       $$
       f(-1) = (-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2
       $$
     - Since $f(-1) \neq 0$, $x + 1$ is not a factor.

- **Option 3: $x + 2$**
     - Substitute $x = -2$ into the polynomial:
       $$
       f(-2) = (-2)^3 - 2(-2) + 1 = -8 + 4 + 1 = -3
       $$
     - Since $f(-2) \neq 0$, $x + 2$ is not a factor.

- **Option 4: $x - 2$**
     - Substitute $x = 2$ into the polynomial:
       $$
       f(2) = 2^3 - 2(2) + 1 = 8 - 4 + 1 = 5
       $$
     - Since $f(2) \neq 0$, $x - 2$ is not a factor.

In conclusion, the only option that satisfies the condition of the Factor Theorem is Option 1, $x - 1$. Therefore, $x - 1$ is the correct factor of the expression $x^3 - 2x + 1$.