Use synthetic division to find the quotient and remainder.
[tex]\frac{4 x^3-39}{x-2}[/tex]
quotient $\square$
remainder $\square$
Answer (2)
Using synthetic division, the quotient of x − 2 4 x 3 − 39 is 4 x 2 + 8 x + 16 and the remainder is − 7 .
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Set up the synthetic division table with the coefficients of the dividend and the root of the divisor.
Perform the synthetic division by bringing down the first coefficient, multiplying by the root, and adding to the next coefficient.
Continue the process until all coefficients have been used.
Identify the quotient and remainder from the resulting numbers: Quotient is 4 x 2 + 8 x + 16 and the remainder is − 7 .
Explanation
Understanding the Problem We are given the division problem x − 2 4 x 3 − 39 . Our goal is to find the quotient and remainder using synthetic division. Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . In this case, a = 2 .
Setting up Synthetic Division First, we set up the synthetic division table. Write down the coefficients of the dividend 4 x 3 − 39 , which are 4, 0, 0, and -39. Note that we include the coefficients of all powers of x , even if they are zero. Write the value of a = 2 to the left.
Performing Synthetic Division Now, we perform the synthetic division:
Bring down the first coefficient (4).
Multiply the root (2) by the brought-down coefficient (4) to get 8, and write it under the second coefficient (0).
Add the second coefficient (0) and the result from the previous step (8) to get 8.
Multiply the root (2) by the result from the previous step (8) to get 16, and write it under the third coefficient (0).
Add the third coefficient (0) and the result from the previous step (16) to get 16.
Multiply the root (2) by the result from the previous step (16) to get 32, and write it under the last coefficient (-39).
Add the last coefficient (-39) and the result from the previous step (32) to get -7. This is the remainder.
Determining the Quotient and Remainder The numbers we obtained in the synthetic division process (4, 8, 16) are the coefficients of the quotient. Since we started with a cubic polynomial ( x 3 ) and divided by a linear factor ( x − 2 ), the quotient will be a quadratic polynomial ( x 2 ). Thus, the quotient is 4 x 2 + 8 x + 16 . The last number, -7, is the remainder.
Final Answer Therefore, the quotient is 4 x 2 + 8 x + 16 and the remainder is -7.
Examples
Synthetic division is a useful tool in various real-world applications, such as polynomial factorization and root finding. For instance, engineers use polynomial models to describe the behavior of physical systems. Synthetic division can help simplify these models, making them easier to analyze and design. Imagine designing a bridge where the load distribution can be modeled by a polynomial. Using synthetic division, engineers can quickly determine critical points and ensure the bridge's stability under different loads. This method streamlines complex calculations, saving time and resources in practical engineering projects.
