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Chapter 2: Problem 18

Use long division to divide. \(\left(x^{3}+125\right) \div(x+5)\)

Short Answer

Expert verified
So, the solution for \((x^{3}+125) \div(x+5)\) using long division is \(x^{2}-5x \) with a remainder of 25.

Step by step solution

01

Align the Dividend and Divisor

First, we set up the long division. Write \(x^{3}+125\) inside the division symbol (or box) and \(x+5\) on the outside.
02

Divide

Next, we divide the first term in the dividend \((x^{3})\) by the first term in the divisor \((x)\). This gives us \(x^{2}\). We write this above the line.
03

Multiply and Subtract

Now we multiply \(x^{2}\), our result from the previous step, by the divisor \(x+5\). This gives us \(x^{3}+5x^{2}\), which we subtract from the dividend, giving us \(-5x^{2}+125\).
04

Repeat the divide, multiply, subtract sequence

Just as in the previous steps, divide the first term of the result \((-5x^{2})\) by the first term of the divisor \((x)\), giving us \(-5x\). Multiplying this by the divisor and subtracting from the above result gives us a final remainder of \(25\). this implies \(-5x^{2}+25\) as our remainder.
05

Finalize the result

The last term in the left-over result \((25)\) is a constant term while the divisor still has \(x\), hence we can't go any further. So the division result is \(x^{2}-5x\) and the remainder is \(25\) after subtracting \(x^{3}+5x^{2}\) from \(x^{3}+125\).

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