Use synthetic division to perform $(x^4-3x+5) \div (x-4)$.
A) $x^3-4x^2+16x+61-249/x-4$
B) $x^3+4x^2+16x+61+249/x-4$
C) $x^3+4x^2+16x+61-249/x-4$
D) $x^3-4x^2+16x+61+249/x-4$
Answer (2)
Set up the synthetic division table with the coefficients of the dividend (1, 0, 0, -3, 5) and the root of the divisor (4).
Perform the synthetic division to find the coefficients of the quotient and the remainder.
The quotient is x 3 + 4 x 2 + 16 x + 61 and the remainder is 249.
Express the result as x 3 + 4 x 2 + 16 x + 61 + x − 4 249 , so the answer is B .
Explanation
Understanding the Problem We are asked to perform synthetic division on the polynomial ( x 4 − 3 x + 5 ) divided by ( x − 4 ) . Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form ( x − a ) .
Setting up Synthetic Division First, we write down the coefficients of the polynomial x 4 − 3 x + 5 . Note that we need to include placeholders for the missing terms x 3 and x 2 . So, the coefficients are 1 , 0 , 0 , − 3 , 5 . We are dividing by x − 4 , so we use 4 as the divisor in the synthetic division.
Performing Synthetic Division Now, we perform the synthetic division:
4 | 1 0 0 -3 5
| 4 16 64 244
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1 4 16 61 249
We bring down the first coefficient (1). Then, we multiply 4 by 1 to get 4, and add it to the next coefficient (0) to get 4. We multiply 4 by 4 to get 16, and add it to the next coefficient (0) to get 16. We multiply 4 by 16 to get 64, and add it to the next coefficient (-3) to get 61. Finally, we multiply 4 by 61 to get 244, and add it to the last coefficient (5) to get 249.
Interpreting the Result The numbers in the bottom row (1, 4, 16, 61) are the coefficients of the quotient, and the last number (249) is the remainder. Since we started with a polynomial of degree 4 and divided by a polynomial of degree 1, the quotient has degree 3. Thus, the quotient is x 3 + 4 x 2 + 16 x + 61 , and the remainder is 249.
Final Result Therefore, the result of the division is x 3 + 4 x 2 + 16 x + 61 + x − 4 249 .
Selecting the Correct Option Comparing this to the given options, we see that the correct answer is B.
Examples
Synthetic division is a useful tool in polynomial algebra, particularly when you need to quickly divide a polynomial by a linear factor. For instance, imagine you are designing a bridge and need to calculate the bending moment along its structure, which can be modeled by a polynomial. If you know a specific point where the bending moment is zero (a root), you can use synthetic division to simplify the polynomial, making it easier to analyze the remaining bending behavior. This simplification helps engineers ensure the bridge's structural integrity and safety.
We performed synthetic division on ( x 4 − 3 x + 5 ) by ( x − 4 ) and found the result to be x 3 + 4 x 2 + 16 x + 61 + x − 4 249 . Therefore, the correct answer is option B. The last number, 249 , is the remainder of the division.
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