What is the result of using synthetic division to divide 3x4−5x3+4x2−7x+1

Question

What is the result of using synthetic division to divide 3x4−5x3+4x2−7x+1
  by x+1
 ?(1 point)
Responses

3x3−8x2+12x−19+20x+1
3 x cubed minus 8 x squared plus 12 x minus 19 plus the fraction with numerator 20 and denominator x plus 1

3x3−8x2+12x−8
3 x cubed minus 8 x squared plus 12 x minus 8

3x3−8x2+12x−8−7x+1
3 x cubed minus 8 x squared plus 12 x minus 8 minus the fraction with numerator 7 and denominator x plus 1

3x3−2x2+2x−5−4x+1

GPT-4o mini · Answer

To use synthetic division to divide $3x^4 - 5x^3 + 4x^2 - 7x + 1$ by $x + 1$, we will substitute $-1$ for $x$ in synthetic division (since we set $x + 1 = 0$ giving $x = -1$). Here are the steps: 1. Write down the coefficients of the polynomial: $3, -5, 4, -7, 1$ 2. Set up synthetic division using $-1$: ``` -1 | 3 -5 4 -7 1 | -3 8 -12 19 ---------------- 3 -8 12 -19 20 ``` 3. Bring down the first coefficient $3$. 4. Multiply $-1$ by $3$ and write the result underneath $-5$: $-1 \times 3 = -3$. 5. Add $-5$ and $-3$ to get $-8$. 6. Repeat the process: - Multiply $-1$ by $-8$ to get $8$, and add to $4$: $4 + 8 = 12$. - Multiply $-1$ by $12$ to get $-12$, and add to $-7$: $-7 - 12 = -19$. - Multiply $-1$ by $-19$ to get $19$, and add to $1$: $1 + 19 = 20$. 7. The result of the synthetic division is therefore: - Quotient: $3x^3 - 8x^2 + 12x - 19$ - Remainder: $20$ So, we can write the result as: $$3x^3 - 8x^2 + 12x - 19 + \frac{20}{x + 1}$$ Thus, the correct response is: **3x^3−8x^2+12x−19+20/(x+1)**.