Use synthetic division to solve $(4 x^3-3 x^2+5 x+6) \div (x+6)$. What is the quotient?
A. $4 x^2-27 x+167-\frac{996}{x-6}$
B. $4 x^2+21 x+131+\frac{792}{x+8}$
C. $4 x^2+21 x+131+\frac{792}{x-6}$
D. $4 x^2-27 x+167-\frac{996}{x+6}$
Answer (1)
Set up synthetic division with the divisor -6 and the coefficients of the polynomial 4, -3, 5, 6.
Perform synthetic division to find the coefficients of the quotient and the remainder.
The coefficients of the quotient are 4, -27, and 167, so the quotient is 4 x 2 − 27 x + 167 .
The remainder is -996, and the result of the division is 4 x 2 − 27 x + 167 − x + 6 996 . The quotient is 4 x 2 − 27 x + 167 .
Explanation
Understanding the Problem We are given a polynomial 4 x 3 − 3 x 2 + 5 x + 6 and we want to divide it by ( x + 6 ) using synthetic division. The goal is to find the quotient.
Setting up Synthetic Division We set up synthetic division with -6 as the divisor and the coefficients of the polynomial as the dividend. The coefficients of the polynomial are 4, -3, 5, 6.
Performing Synthetic Division We perform synthetic division:
Bring down the first coefficient (4).
Multiply the divisor (-6) by the first coefficient (4) and write the result (-24) under the second coefficient (-3).
Add the second coefficient (-3) and the result (-24) to get -27.
Multiply the divisor (-6) by -27 and write the result (162) under the third coefficient (5).
Add the third coefficient (5) and the result (162) to get 167.
Multiply the divisor (-6) by 167 and write the result (-1002) under the last coefficient (6).
Add the last coefficient (6) and the result (-1002) to get -996. This is the remainder.
Determining the Quotient and Remainder The coefficients of the quotient are 4, -27, and 167. The quotient is 4 x 2 − 27 x + 167 . The remainder is -996. So the result of the division is 4 x 2 − 27 x + 167 − x + 6 996 .
Final Answer The quotient is 4 x 2 − 27 x + 167 .
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 along its length, you might model the bending moment as a polynomial function. Using synthetic division, you can quickly determine how the bending moment changes as you move along the bridge, helping you ensure the structure's stability.
