Use synthetic division to find the result when \( 3 x^{4}-19 x^{3}+30 x^{2}-2 x+11 \) is divided by \( x-3 \). If there is a remainder, express the result in the form \( q(x)+\frac{r( }{b(x} \)

To divide the polynomial \( 3x^{4} - 19x^{3} + 30x^{2} - 2x + 11 \) by \( x - 3 \) using synthetic division, follow these steps: ### Step 1: Set Up Synthetic Division 1. **Identify \( c \):** Since we are dividing by \( x - 3 \), \( c = 3 \). 2. **Write Down the Coefficients:** List the coefficients of the dividend polynomial in descending order of degree: \[ 3 \quad -19 \quad 30 \quad -2 \quad 11 \] ### Step 2: Perform Synthetic Division \[ \begin{array}{c|ccccc} 3 & 3 & -19 & 30 & -2 & 11 \\ \hline & & 9 & -30 & 0 & -6 \\ \hline & 3 & -10 & 0 & -2 & 5 \\ \end{array} \] **Explanation of Each Step:** 1. **Bring Down the First Coefficient:** - Bring down the **3**. 2. **Multiply and Add:** - Multiply **3** (the value just brought down) by **3** (our \( c \)) to get **9**. - Add **-19** (the next coefficient) and **9** to get **-10**. 3. **Repeat the Process:** - Multiply **-10** by **3** to get **-30**. - Add **30** and **-30** to get **0**. - Multiply **0** by **3** to get **0**. - Add **-2** and **0** to get **-2**. - Multiply **-2** by **3** to get **-6**. - Add **11** and **-6** to get **5**. ### Step 3: Interpret the Result The bottom row of the synthetic division (excluding the last number) represents the coefficients of the quotient polynomial, and the last number is the remainder. - **Quotient Coefficients:** \( 3 \), \( -10 \), \( 0 \), \( -2 \) - **Remainder:** \( 5 \) Thus, the quotient polynomial is: \[ 3x^{3} - 10x^{2} - 2 \] ### Step 4: Express the Final Result Including the remainder, the division can be expressed as: \[ \frac{3x^{4} - 19x^{3} + 30x^{2} - 2x + 11}{x - 3} = 3x^{3} - 10x^{2} - 2 + \frac{5}{x - 3} \] **Final Answer:** \[ 3x^{3} - 10x^{2} - 2 + \frac{5}{x - 3} \]