Use synthetic division to find the quotient and remainder when $-x^3+4 x^2-8 x+2$ is divided by $x-2$ by completing the parts below.
(a) Complete this synthetic division table.
2) $\begin{array}{llll}-1 & 4 & -8 & 2\end{array}$
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Answer (2)
Bring down the first coefficient: -1.
Multiply and add: 2 × − 1 = − 2 , 4 + ( − 2 ) = 2 .
Multiply and add: 2 × 2 = 4 , − 8 + 4 = − 4 .
Multiply and add: 2 × − 4 = − 8 , 2 + ( − 8 ) = − 6 . The remainder is -6. − 2 , 2 , 4 , − 4 , − 8 , − 6
Explanation
Understanding the Problem We are asked to use synthetic division to divide the polynomial − x 3 + 4 x 2 − 8 x + 2 by x − 2 . We need to complete the synthetic division table. The divisor is x − 2 , so we use 2 in the synthetic division. The coefficients of the dividend − x 3 + 4 x 2 − 8 x + 2 are -1, 4, -8, and 2.
Step 1: Bring down the first coefficient First, write down the coefficients of the polynomial − x 3 + 4 x 2 − 8 x + 2 , which are -1, 4, -8, and 2. Bring down the first coefficient, -1.
Step 2: Multiply and Add (First Iteration) Multiply -1 by 2 to get -2. Add -2 to 4 to get 2.
Step 3: Multiply and Add (Second Iteration) Multiply 2 by 2 to get 4. Add 4 to -8 to get -4.
Step 4: Multiply and Add (Third Iteration) Multiply -4 by 2 to get -8. Add -8 to 2 to get -6.
Final Result The numbers -1, 2, and -4 are the coefficients of the quotient, and -6 is the remainder. The quotient is − x 2 + 2 x − 4 and the remainder is -6. The completed synthetic division table is:
− 1 h l in e − 1 4 2 − 8 − 4 2 − 6 − 2 4 − 8
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 end up with a polynomial equation. Using synthetic division, you can quickly find the roots of the polynomial, which can help you determine the critical points where the bending moment is maximum or minimum, ensuring the bridge's structural integrity.
By using synthetic division, we divided the polynomial − x 3 + 4 x 2 − 8 x + 2 by x − 2 and found the quotient to be − x 2 + 2 x − 4 with a remainder of − 6 . The process involved bringing down coefficients, multiplying, and adding sequentially. The completed synthetic division table outlined the steps of this calculation clearly.
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