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Chapter 3: Problem 57

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

Short Answer

Expert verified
The result of the division is the polynomial \(2x^2+3x-2\) with a remainder of 1. Thus, \(2 x^{3}-3 x^{2}-11 x+7 = (x - 3)(2x^2 + 3x - 2) + 1\).

Step by step solution

01

Setup the Long Division

Write the two polynomials as you would in long division, with \(2 x^{3}-3 x^{2}-11 x+7\) inside and \(x-3\) outside the division bar.
02

Divide the First Term

Divide the first term of the dividend (the inside polynomial) \(2x^3\) by the first term of the divisor (the outside polynomial) \(x\). This gives \(2x^2\), which is the first term of the quotient. Write this term above the long division bar.
03

Multiply the Divisor by the new Quotient

Multiply the divisor \(x-3\) by the new term in the quotient (\(2x^2\)). Write the result \(- 6x^{2} + 6x\) under the first two terms of the dividend.
04

Subtract

Subtract the expression \(6x^{2} - 6x\) obtained in step 3 from the first two terms of the dividend \(2x^{3}-3x^{2}\). You'll be left with \(3x^{2}\).
05

Repeat the Process

Bring down the next term from the dividend (-11x) to form \(3x^{2} - 11x\), then divide, multiply, and add, as per steps 2-4. The new quotient term is \(3x\) derived from \((3x^{2})/x\). So subtract \((3x^2-9x)\) from \((3x^2-11x)\) to give \(-2x\).
06

Finish the Division

Repeat the process with the remaining terms. Bring down the final term (+7) to give \(-2x + 7\), the new quotient term is -2 derived from \((-2x)/x\). So, subtract \(-2x+6\) from \(-2x+7\) to get 1, which is the remainder after the division.

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