Translation of 6 units to the right and 3 units down, translation of 6 units to the right and 3 units up, translation of 6 units to the left and 3 units up.
[tex]m[/tex] the division: [tex]\frac{x^4-7 x^3-14 x^2-11 x+4}{x+1}[/tex]
[tex]\begin{array}{l}
3-8 x^2-6 x-5+\frac{9}{x+1} \\
x^3-6 x^2-20 x-31-\frac{27}{x-1}
\end{array}[/tex]
B) [tex]x^3-8[/tex]
D) [tex]x^3-[/tex]
Answer (2)
Perform polynomial long division of x + 1 x 4 − 7 x 3 − 14 x 2 − 11 x + 4 .
The result of the long division is x 3 − 8 x 2 − 6 x − 5 + x + 1 9 .
Compare the result with the given options.
The correct answer is x 3 − 8 x 2 − 6 x − 5 + x + 1 9 .
Explanation
Problem Analysis We are given the polynomial division problem: x + 1 x 4 − 7 x 3 − 14 x 2 − 11 x + 4 . Our goal is to find the quotient and remainder of this division.
Solution Strategy We will perform polynomial long division to find the quotient and remainder.
Polynomial Long Division Performing the polynomial long division of ( x 4 − 7 x 3 − 14 x 2 − 11 x + 4 ) by ( x + 1 ) , we obtain the quotient x 3 − 8 x 2 − 6 x − 5 and the remainder 9 . This can be written as: x + 1 x 4 − 7 x 3 − 14 x 2 − 11 x + 4 = x 3 − 8 x 2 − 6 x − 5 + x + 1 9
Comparison with Options Comparing our result with the given options, we see that it matches the first option: x 3 − 8 x 2 − 6 x − 5 + x + 1 9
Final Answer Therefore, the correct result of the polynomial division is x 3 − 8 x 2 − 6 x − 5 + x + 1 9 .
Examples
Polynomial division is a fundamental concept in algebra with various real-world applications. For example, engineers use polynomial division to analyze the stability of control systems. Imagine designing a cruise control system for a car; engineers use polynomials to model the system's behavior and polynomial division to simplify these models, ensuring the car maintains a steady speed without unwanted oscillations. This process helps in optimizing the system's performance and safety.
To find the quotient of x + 1 x 4 − 7 x 3 − 14 x 2 − 11 x + 4 , we perform polynomial long division, resulting in x 3 − 8 x 2 − 6 x − 5 + x + 1 9 . The correct answer matches the first given choice. Thus, the answer is x 3 − 8 x 2 − 6 x − 5 + x + 1 9 .
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