Divide the polynomial $3x^2 - 4x^3-3x - 1$ by $x-1$ the remainder is

Q: Divide the polynomial $3x^2 - 4x^3-3x - 1$ by $x-1$ the remainder is

To determine the remainder when the polynomial $3x^2 - 4x^3 - 3x - 1$ is divided by $x - 1$, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear divisor $x - c$ is $f(c)$. In this case, the divisor is $x - 1$, so we need to evaluate the polynomial at $x = 1$: \[ f(x) = -4x^3 + 3x^2 - 3x - 1 \] Substitute $x = 1$ into the polynomial: \[ f(1) = -4(1)^3 + 3(1)^2 - 3(1) - 1 \] Calculate each term: \[ = -4(1) + 3(1) - 3 - 1 \] \[ = -4 + 3 - 3 - 1 \] \[ = -5 \] Thus, the remainder when $3x^2 - 4x^3 - 3x - 1$ is divided by $x - 1$ is $-5$. ### Explanation of Options: 1. **Option 1: -5** This is the correct answer because, as calculated above, substituting $x = 1$ into the polynomial results in a remainder of $-5$. 2. **Option 2: 5** This option is incorrect because the calculation of the remainder using the Remainder Theorem gives $-5$, not $5$. 3. **Option 3: 2** This option is incorrect because substituting $x = 1$ into the polynomial does not yield a remainder of $2$. The correct remainder is $-5$. 4. **Option 4: -2** This option is incorrect because the remainder calculated using the Remainder Theorem is $-5$, not $-2$. In conclusion, the correct remainder when dividing the polynomial by $x - 1$ is $-5$, which corresponds to Option 1.

Question

Accepted Answer

To determine the remainder when the polynomial $3x^2 - 4x^3 - 3x - 1$ is divided by $x - 1$, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear divisor $x - c$ is $f(c)$.

In this case, the divisor is $x - 1$, so we need to evaluate the polynomial at $x = 1$:

$$ f(x) = -4x^3 + 3x^2 - 3x - 1 $$

Substitute $x = 1$ into the polynomial:

$$
f(1) = -4(1)^3 + 3(1)^2 - 3(1) - 1
$$

Calculate each term:

$$
= -4(1) + 3(1) - 3 - 1
$$

$$
= -4 + 3 - 3 - 1
$$

$$
= -5
$$

Thus, the remainder when $3x^2 - 4x^3 - 3x - 1$ is divided by $x - 1$ is $-5$.

### Explanation of Options:

1. **Option 1: -5**  
   This is the correct answer because, as calculated above, substituting $x = 1$ into the polynomial results in a remainder of $-5$.

2. **Option 2: 5**  
   This option is incorrect because the calculation of the remainder using the Remainder Theorem gives $-5$, not $5$.

3. **Option 3: 2**  
   This option is incorrect because substituting $x = 1$ into the polynomial does not yield a remainder of $2$. The correct remainder is $-5$.

4. **Option 4: -2**  
   This option is incorrect because the remainder calculated using the Remainder Theorem is $-5$, not $-2$.

In conclusion, the correct remainder when dividing the polynomial by $x - 1$ is $-5$, which corresponds to Option 1.