Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\left(x^{3}-9\right) \div\left(x^{2}+1\right)$$

Step-by-Step Solution

Answer

The quotient is \(x\), and the remainder is \(-x^{2}-9\). So, the division \((x^{3}-9) \div (x^{2}+1)\) is written as \(x - \frac{x^{2}+9}{x^{2}+1}\).

Step 1: Match the highest degree terms

First match the highest degree term of the divisor \(x^{2}+1\) with the highest degree term of the dividend \(x^{3}-9\). Corresponding to \(x^{3}\), we select \(x\). We multiply entire divisor with \(x\) to match the highest degree term. Therefore, \(x\) is the first term of our quotient.

Step 2: Subtract from the dividend

Now multiply the divisor by the term obtained above and subtract the result from the original dividend. i.e., \((x^{3}-9) - x\cdot(x^{2}+1) = -x^{2}-9\). This is our new dividend.

Step 3: Repeat the process

Now we repeat the process. We can't match \(x^{2}\) with the highest degree term of \(-x^{2}-9\), as degree of \(-x^{2}-9\) is lesser. Thus, we stop here.

Step 4: Write remaining dividend as the remainder

We write the remaining dividend as remainder. Here the remainder is \(-x^{2}-9\).