The quotient of $(x^4-3 x^2+4 x-3)$ and $(x^2+x-3)$ is a polynomial. What is the quotient?
Answer (2)
Perform polynomial long division of x 4 − 3 x 2 + 4 x − 3 by x 2 + x − 3 .
The quotient obtained from the long division is x 2 − x + 1 .
Verify the result by multiplying the quotient with the divisor x 2 + x − 3 to ensure it equals x 4 − 3 x 2 + 4 x − 3 .
The quotient is x 2 − x + 1 .
Explanation
Problem Analysis We are given two polynomials: x 4 − 3 x 2 + 4 x − 3 and x 2 + x − 3 . Our goal is to find the quotient when the first polynomial is divided by the second.
Solution Strategy We will perform polynomial long division to find the quotient.
Polynomial Long Division Performing polynomial long division of x 4 − 3 x 2 + 4 x − 3 by x 2 + x − 3 :
\multicolumn 2 r x 2 \cline 2 − 5 x 2 + x − 3 \multicolumn 2 r x 4 \cline 2 − 4 \multicolumn 2 r 0 \multicolumn 2 r \cline 3 − 5 \multicolumn 2 r \multicolumn 2 r \cline 4 − 6 \multicolumn 2 r − x x 4 + x 3 − x 3 − x 3 0 + 1 + 0 x 3 − 3 x 2 + 0 x 2 − x 2 x 2 x 2 0 − 3 x 2 + 4 x + 3 x + x + x 0 + 4 x − 3 − 3 0 − 3
The quotient is x 2 − x + 1 .
Verification To verify the result, we multiply the quotient ( x 2 − x + 1 ) by the divisor ( x 2 + x − 3 ) :
( x 2 − x + 1 ) ( x 2 + x − 3 ) = x 4 + x 3 − 3 x 2 − x 3 − x 2 + 3 x + x 2 + x − 3 = x 4 − 3 x 2 + 4 x − 3
This matches the original polynomial, so our quotient is correct.
Final Answer The quotient of the given polynomials is x 2 − x + 1 .
Examples
Polynomial division is used in various engineering and scientific applications. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Simplifying this ratio using polynomial division can help in analyzing the system's stability and response. Similarly, in signal processing, polynomial division can be used to decode error-correcting codes.
The quotient of the polynomial x 4 − 3 x 2 + 4 x − 3 when divided by x 2 + x − 3 is x 2 − x + 1 . This was found through polynomial long division and verified by multiplication. The steps were performed systematically to ensure accuracy.
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