Select the correct answer. Using synthetic division, find $(2 x^4+4 x^3+2 x^2+8 x+8) \div(x+2)$. A. $2 x^3+2 x+4$ B. $2 x^4+2 x^2+4 x$ C. $2 x^3+2 x+4+\frac{1}{x+2}$ D. $2 x^3+2 x^2+4
Answer (2)
Set up the synthetic division with the coefficients of the dividend and the value from the divisor.
Perform the synthetic division to find the coefficients of the quotient and the remainder.
Write the quotient polynomial using the coefficients obtained.
The quotient is 2 x 3 + 2 x + 4 .
Explanation
Understanding the Problem We are given the polynomial division problem ( 2 x 4 + 4 x 3 + 2 x 2 + 8 x + 8 ) ÷ ( x + 2 ) and asked to use synthetic division to find the quotient.
Setting up Synthetic Division We set up the synthetic division table with the coefficients of the dividend (2, 4, 2, 8, 8) and the value c = − 2 from the divisor x + 2 = x − ( − 2 ) .
Performing Synthetic Division Performing synthetic division:
Bring down the first coefficient: 2
Multiply by c = − 2 : 2 × − 2 = − 4
Add to the next coefficient: 4 + ( − 4 ) = 0
Multiply by c = − 2 : 0 × − 2 = 0
Add to the next coefficient: 2 + 0 = 2
Multiply by c = − 2 : 2 × − 2 = − 4
Add to the next coefficient: 8 + ( − 4 ) = 4
Multiply by c = − 2 : 4 × − 2 = − 8
Add to the next coefficient: 8 + ( − 8 ) = 0
The coefficients of the quotient are 2, 0, 2, and 4, and the remainder is 0.
Writing the Quotient The degree of the quotient will be one less than the degree of the dividend. Since the dividend has degree 4, the quotient has degree 3. Thus, the quotient is 2 x 3 + 0 x 2 + 2 x + 4 = 2 x 3 + 2 x + 4 .
Selecting the Correct Answer Comparing the resulting quotient with the given options, we find that the correct answer is 2 x 3 + 2 x + 4 .
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c . It's useful in various applications, such as finding roots of polynomials, simplifying expressions, and solving problems in calculus. For example, engineers might use synthetic division to analyze the stability of a system modeled by a polynomial equation. By finding the roots of the polynomial, they can determine whether the system will remain stable over time. This technique is also used in computer graphics for tasks like curve fitting and surface modeling, where polynomials are used to represent shapes and trajectories.
Using synthetic division on ( 2 x 4 + 4 x 3 + 2 x 2 + 8 x + 8 ) ÷ ( x + 2 ) , we find the quotient to be 2 x 3 + 2 x + 4 . Hence, the answer is option C. This illustrates a practical application of synthetic division in polynomial expressions.
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