Question

1. $(x^5 - 2x^4 + x^3 - 8x + 18) div (x^2 - 3)$

Explanation:

Step1: Perform Polynomial Long Division

Divide the leading term of the dividend \(x^5 - 2x^4 + x^3 - 8x + 18\) by the leading term of the divisor \(x^2 - 3\). The leading term of the dividend is \(x^5\) and the leading term of the divisor is \(x^2\), so \(x^5\div x^2 = x^3\). Multiply the divisor \(x^2 - 3\) by \(x^3\) to get \(x^5 - 3x^3\). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step2: Divide the New Polynomial

Now, divide the leading term of \(-2x^4 + 4x^3 - 8x + 18\) (which is \(-2x^4\)) by the leading term of the divisor (\(x^2\)) to get \(-2x^2\). Multiply the divisor \(x^2 - 3\) by \(-2x^2\) to get \(-2x^4 + 6x^2\). Subtract this from the polynomial:

$$
LATEXBLOCK1
$$

Step3: Divide the New Polynomial Again

Divide the leading term of \(4x^3 - 6x^2 - 8x + 18\) (which is \(4x^3\)) by the leading term of the divisor (\(x^2\)) to get \(4x\). Multiply the divisor \(x^2 - 3\) by \(4x\) to get \(4x^3 - 12x\). Subtract this from the polynomial:

$$
LATEXBLOCK2
$$

Step4: Divide the New Polynomial Once More

Divide the leading term of \(-6x^2 + 4x + 18\) (which is \(-6x^2\)) by the leading term of the divisor (\(x^2\)) to get \(-6\). Multiply the divisor \(x^2 - 3\) by \(-6\) to get \(-6x^2 + 18\). Subtract this from the polynomial:

$$
LATEXBLOCK3
$$

Since the degree of the remainder \(4x\) is less than the degree of the divisor \(x^2 - 3\), we stop here.

The quotient is \(x^3 - 2x^2 + 4x - 6\) and the remainder is \(4x\). So, \((x^5 - 2x^4 + x^3 - 8x + 18)\div(x^2 - 3)=x^3 - 2x^2 + 4x - 6+\frac{4x}{x^2 - 3}\)

Answer:

The quotient is \(x^3 - 2x^2 + 4x - 6\) and the remainder is \(4x\), so the result of the division is \(x^3 - 2x^2 + 4x - 6 + \frac{4x}{x^2 - 3}\)