Find the quotient using long division.
$\frac{3 x^3+3 x^2-21 x+3}{x-2}$
Answer (2)
Using long division, we find that the quotient of x − 2 3 x 3 + 3 x 2 − 21 x + 3 is 3 x 2 + 9 x − 3 with a remainder of − 3 . The complete result can be expressed as 3 x 2 + 9 x − 3 + x − 2 − 3 .
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Set up the long division with dividend 3 x 3 + 3 x 2 − 21 x + 3 and divisor x − 2 .
Perform the long division step by step, dividing, multiplying, and subtracting.
Identify the quotient and the remainder.
Express the result as the quotient plus the remainder divided by the divisor: 3 x 2 + 9 x − 3 + x − 2 − 3 . The quotient is 3 x 2 + 9 x − 3 .
Explanation
Understanding the problem We are asked to find the quotient of the division x − 2 3 x 3 + 3 x 2 − 21 x + 3 using long division. Long division is a method for dividing polynomials, similar to how we divide numbers.
Performing long division First, set up the long division problem with the dividend ( 3 x 3 + 3 x 2 − 21 x + 3 ) and the divisor ( x − 2 ).
\multicolumn 2 r \cline 2 − 5 x − 2 \multicolumn 2 r 3 x 3 \cline 2 − 3 \multicolumn 2 r 0 \multicolumn 2 r \cline 3 − 4 \multicolumn 2 r \multicolumn 2 r \cline 4 − 5 \multicolumn 2 r 3 x 2 3 x 3 − 6 x 2 9 x 2 9 x 2 0 + 9 x + 3 x 2 − 21 x − 18 x − 3 x − 3 x 0 − 3 − 21 x + 3 + 6 − 3 + 3
Detailed calculations Divide the leading term of the dividend ( 3 x 3 ) by the leading term of the divisor ( x ) to get the first term of the quotient ( 3 x 2 ). Multiply the divisor ( x − 2 ) by the first term of the quotient ( 3 x 2 ) to get 3 x 3 − 6 x 2 .
Subtract this result ( 3 x 3 − 6 x 2 ) from the dividend ( 3 x 3 + 3 x 2 − 21 x + 3 ) to get the new dividend ( 9 x 2 − 21 x + 3 ). Divide the leading term of the new dividend ( 9 x 2 ) by the leading term of the divisor ( x ) to get the next term of the quotient ( 9 x ). Multiply the divisor ( x − 2 ) by this term ( 9 x ) to get 9 x 2 − 18 x .
Subtract this result ( 9 x 2 − 18 x ) from the new dividend ( 9 x 2 − 21 x + 3 ) to get the next new dividend ( − 3 x + 3 ). Divide the leading term of the new dividend ( − 3 x ) by the leading term of the divisor ( x ) to get the next term of the quotient ( − 3 ). Multiply the divisor ( x − 2 ) by this term ( − 3 ) to get − 3 x + 6 .
Subtract this result ( − 3 x + 6 ) from the new dividend ( − 3 x + 3 ) to get the remainder ( − 3 ).
Final result Write the quotient as the sum of the terms obtained in steps 2, 5, and 8, plus the remainder divided by the divisor: 3 x 2 + 9 x − 3 + x − 2 − 3 .
Therefore, the quotient is 3 x 2 + 9 x − 3 with a remainder of − 3 .
Examples
Polynomial long division is used in various engineering and scientific applications, such as control systems design, signal processing, and structural analysis. For instance, in control systems, engineers use polynomial division to simplify transfer functions, which describe the relationship between the input and output of a system. By dividing polynomials, they can reduce the complexity of the transfer function, making it easier to analyze and design controllers that ensure the system operates efficiently and stably. This technique helps in optimizing the performance of systems ranging from aircraft autopilots to industrial robots.
