Divide using synthetic division.
$\left(x^4+11 x^3+29 x^2-8 x-48\right) \div(x+4)$
Answer (1)
Set up the synthetic division table with the coefficients of the dividend and the divisor.
Perform the synthetic division steps: bring down, multiply, and add.
Read the coefficients of the quotient and the remainder from the bottom row.
The quotient is x 3 + 7 x 2 + x − 12 . x 3 + 7 x 2 + x − 12
Explanation
Understanding the Problem We are asked to divide the polynomial x 4 + 11 x 3 + 29 x 2 − 8 x − 48 by ( x + 4 ) using synthetic division. Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c . In this case, c = − 4 .
Setting up Synthetic Division First, we set up the synthetic division table. Write down the coefficients of the polynomial: 1, 11, 29, -8, -48. Write the value of c = − 4 to the left.
Step 1: Bring Down Bring down the first coefficient (1) to the bottom row.
Step 2: Multiply and Add Multiply -4 by 1 to get -4. Write -4 under the second coefficient (11). Add 11 and -4 to get 7.
Step 3: Multiply and Add Multiply -4 by 7 to get -28. Write -28 under the third coefficient (29). Add 29 and -28 to get 1.
Step 4: Multiply and Add Multiply -4 by 1 to get -4. Write -4 under the fourth coefficient (-8). Add -8 and -4 to get -12.
Step 5: Multiply and Add Multiply -4 by -12 to get 48. Write 48 under the fifth coefficient (-48). Add -48 and 48 to get 0.
Result The numbers in the bottom row are 1, 7, 1, -12, and 0. These are the coefficients of the quotient and the remainder. The quotient is x 3 + 7 x 2 + x − 12 , and the remainder is 0.
Final Answer Therefore, ( x 4 + 11 x 3 + 29 x 2 − 8 x − 48 ) ÷ ( x + 4 ) = x 3 + 7 x 2 + x − 12 .
Examples
Synthetic division is a useful tool in various real-world applications, such as polynomial modeling in engineering and physics. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. If they need to adjust the design by changing a parameter, synthetic division can help quickly determine how the load distribution polynomial changes, ensuring the bridge's stability and safety. This method simplifies complex calculations, making it easier to analyze and optimize structural designs.
