Now we will set up the Synthetic Division:
8. What is the solution after we perform the synthetic division?
Quotient = $\square$
Answer (1)
Complete the synthetic division.
Identify the coefficients of the quotient.
Write the quotient as a polynomial: 2 x + 5 .
The quotient is 2 x + 5 .
Explanation
Understanding the Problem The problem presents a partially completed synthetic division and asks for the quotient. We need to complete the synthetic division and identify the quotient.
Completing the Synthetic Division The synthetic division is set up to divide a polynomial by ( x − 4 ) . The coefficients of the polynomial being divided are 2, -3, and -20. The setup looks like this:
4 | 2 -3 -20
|
First, bring down the leading coefficient (2). Then multiply 4 by 2 to get 8, and add it to -3 to get 5. Then multiply 4 by 5 to get 20, and add it to -20 to get 0.
4 | 2 -3 -20
| 8 20
2 5 0
The numbers 2 and 5 are the coefficients of the quotient, and 0 is the remainder.
Identifying the Quotient The quotient is therefore 2 x + 5 .
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - a. It's often used in algebra to find the roots of polynomials or to simplify expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might model the bending moment as a polynomial function. Using synthetic division, you can quickly determine if a particular support location (represented by 'a') will result in zero bending moment, indicating a stable point. This helps engineers optimize the placement of supports to ensure the bridge's structural integrity.
