Use synthetic division to find the quotient and remainder. (Simplify your answers completely.)
2×3 − 11×2 + 18x − 17 divided by x − 4
The Correct Answer and Explanation is:
To solve the problem ( \frac{2x^3 – 11x^2 + 18x – 17}{x – 4} ) using synthetic division, we follow these steps:
Step 1: Set up the synthetic division table
We need to divide ( 2x^3 – 11x^2 + 18x – 17 ) by ( x – 4 ). Since we are dividing by ( x – 4 ), we will use ( 4 ) as the divisor in the synthetic division process. We list the coefficients of the polynomial ( 2x^3 – 11x^2 + 18x – 17 ), which are ( 2, -11, 18, -17 ).
The synthetic division setup is as follows:
[
\begin{array}{r|rrrr}
4 & 2 & -11 & 18 & -17 \
& & 8 & -12 & 24 \
\hline
& 2 & -3 & 6 & 7 \
\end{array}
]
Step 2: Perform synthetic division
- Bring down the first coefficient (2).
- Multiply 2 (the number you brought down) by 4 (the divisor) and write the result under the second coefficient: ( 2 \times 4 = 8 ).
- Add this result to the second coefficient: ( -11 + 8 = -3 ).
- Multiply ( -3 ) by 4: ( -3 \times 4 = -12 ), and add this to the next coefficient: ( 18 + (-12) = 6 ).
- Multiply 6 by 4: ( 6 \times 4 = 24 ), and add this to the final coefficient: ( -17 + 24 = 7 ).
Step 3: Interpret the result
From the synthetic division table, we see that the quotient is ( 2x^2 – 3x + 6 ) and the remainder is ( 7 ).
Thus, the division of ( 2x^3 – 11x^2 + 18x – 17 ) by ( x – 4 ) yields:
[
\frac{2x^3 – 11x^2 + 18x – 17}{x – 4} = 2x^2 – 3x + 6 + \frac{7}{x – 4}
]
Final Answer:
- Quotient: ( 2x^2 – 3x + 6 )
- Remainder: ( 7 )
Explanation:
Synthetic division is a simplified method for dividing polynomials when the divisor is a linear binomial of the form ( x – c ). This method eliminates the need for long division and is especially useful for polynomials with higher degrees. It involves writing down the coefficients of the dividend polynomial and using the value of ( c ) from ( x – c ) (in this case, ( c = 4 )) to perform the division step-by-step, multiplying and adding the terms as shown.