Question

1. $(x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}$

Explanation:

Step1: Rewrite the expression

The expression \((x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}\) can be rewritten as \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}\) using the property \(a^{-n}=\frac{1}{a^{n}}\).

Step2: Perform polynomial long division

We divide the polynomial \(x^{4}-3x^{3}+0x^{2}+5x - 6\) (we added \(0x^{2}\) to make the polynomial complete for long division) by \(x + 2\).

• Divide the leading term \(x^{4}\) by \(x\) to get \(x^{3}\). Multiply \(x + 2\) by \(x^{3}\) to get \(x^{4}+2x^{3}\). Subtract this from the dividend:

\((x^{4}-3x^{3}+0x^{2}+5x - 6)-(x^{4}+2x^{3})=- 5x^{3}+0x^{2}+5x - 6\)

• Divide the leading term \(-5x^{3}\) by \(x\) to get \(-5x^{2}\). Multiply \(x + 2\) by \(-5x^{2}\) to get \(-5x^{3}-10x^{2}\). Subtract this from the previous remainder:

\((-5x^{3}+0x^{2}+5x - 6)-(-5x^{3}-10x^{2}) = 10x^{2}+5x - 6\)

• Divide the leading term \(10x^{2}\) by \(x\) to get \(10x\). Multiply \(x + 2\) by \(10x\) to get \(10x^{2}+20x\). Subtract this from the previous remainder:

\((10x^{2}+5x - 6)-(10x^{2}+20x)=-15x - 6\)

• Divide the leading term \(-15x\) by \(x\) to get \(-15\). Multiply \(x + 2\) by \(-15\) to get \(-15x-30\). Subtract this from the previous remainder:

\((-15x - 6)-(-15x - 30)=24\)

So, \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}=x^{3}-5x^{2}+10x - 15+\frac{24}{x + 2}\)

Answer:

\(x^{3}-5x^{2}+10x - 15+\frac{24}{x + 2}\)