What is the quotient of [tex]$\left(2 x^4-3 x^3-3 x^2+7 x-3\right) \div\left(x^2-2 x+1\right)$[/tex]?
Answer (1)
Perform polynomial long division of ( 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 ) by ( x 2 − 2 x + 1 ) .
The quotient is found to be 2 x 2 + x − 3 .
Therefore, the answer is 2 x 2 + x − 3 .
Explanation
Problem Analysis We are given the polynomial division problem: ( 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 ) ÷ ( x 2 − 2 x + 1 ) . We need to find the quotient.
Solution Strategy We can perform polynomial long division to find the quotient. Alternatively, we can try to factor the dividend and see if the divisor is a factor.
Polynomial Division Performing the polynomial division, we have: ( 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 ) ÷ ( x 2 − 2 x + 1 ) = 2 x 2 + x − 3
Final Result Therefore, the quotient is 2 x 2 + x − 3 .
Examples
Polynomial division is a fundamental concept in algebra with numerous real-world applications. For instance, engineers use polynomial division to analyze and design control systems, ensuring stability and desired performance. In economics, polynomial division can be used to model and predict market trends, helping businesses make informed decisions about production and pricing. Moreover, in computer graphics, polynomial division plays a crucial role in creating smooth curves and surfaces, enhancing the visual appeal and realism of virtual environments.
