Question

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

Explanation:

Step1: Simplify the negative exponent

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \((x + 2)^{-1}=\frac{1}{x + 2}\). The expression is \((x^{4}-3x^{3}+5x - 6)\times\frac{1}{x + 2}\), which is equivalent to \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}\). We can use polynomial long division or synthetic division. Let's use polynomial long division.
Divide \(x^{4}-3x^{3}+0x^{2}+5x - 6\) by \(x + 2\).

Step2: Divide the leading terms

Divide \(x^{4}\) by \(x\) to get \(x^{3}\). Multiply \(x + 2\) by \(x^{3}\) to get \(x^{4}+2x^{3}\). Subtract this from the dividend: \((x^{4}-3x^{3}+0x^{2}+5x - 6)-(x^{4}+2x^{3})=-5x^{3}+0x^{2}+5x - 6\).

Step3: Divide the new leading term

Divide \(-5x^{3}\) by \(x\) to get \(-5x^{2}\). Multiply \(x + 2\) by \(-5x^{2}\) to get \(-5x^{3}-10x^{2}\). Subtract: \((-5x^{3}+0x^{2}+5x - 6)-(-5x^{3}-10x^{2}) = 10x^{2}+5x - 6\).

Step4: Divide the new leading term

Divide \(10x^{2}\) by \(x\) to get \(10x\). Multiply \(x + 2\) by \(10x\) to get \(10x^{2}+20x\). Subtract: \((10x^{2}+5x - 6)-(10x^{2}+20x)=-15x - 6\).

Step5: Divide the new leading term

Divide \(-15x\) by \(x\) to get \(-15\). Multiply \(x + 2\) by \(-15\) to get \(-15x-30\). Subtract: \((-15x - 6)-(-15x - 30)=24\).
So, \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}=x^{3}-5x^{2}+10x - 15+\frac{24}{x + 2}\) (if we consider the division with remainder) or if we made a mistake in the problem (maybe it's a typo and the original expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)\) instead of with the negative exponent? Wait, the original expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}\), but maybe there was a mis - writing. Wait, if we assume that the underlined part is a mistake and the expression is \((x^{4}-3x^{3}+5x - 6)\) divided by \((x + 2)^{-1}\), which is \((x^{4}-3x^{3}+5x - 6)(x + 2)\). Let's recalculate that.
If the expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)\), then:
Multiply \(x^{4}\) by \(x\) and \(2\): \(x^{5}+2x^{4}\)
Multiply \(-3x^{3}\) by \(x\) and \(2\): \(-3x^{4}-6x^{3}\)
Multiply \(0x^{2}\) by \(x\) and \(2\): \(0x^{3}+0x^{2}\)
Multiply \(5x\) by \(x\) and \(2\): \(5x^{2}+10x\)
Multiply \(-6\) by \(x\) and \(2\): \(-6x-12\)
Now, combine like terms:
\(x^{5}+(2x^{4}-3x^{4})+(-6x^{3}+0x^{3})+(0x^{2}+5x^{2})+(10x - 6x)+(-12)\)
\(=x^{5}-x^{4}-6x^{3}+5x^{2}+4x - 12\)
Wait, the original problem has a fraction - like underlining, maybe the expression is \(\frac{(x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}}{(x + 2)^{-1}}\). If that's the case,[SSE onError error]

Answer:

Step1: Simplify the negative exponent

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \((x + 2)^{-1}=\frac{1}{x + 2}\). The expression is \((x^{4}-3x^{3}+5x - 6)\times\frac{1}{x + 2}\), which is equivalent to \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}\). We can use polynomial long division or synthetic division. Let's use polynomial long division.
Divide \(x^{4}-3x^{3}+0x^{2}+5x - 6\) by \(x + 2\).

Step2: Divide the leading terms

Divide \(x^{4}\) by \(x\) to get \(x^{3}\). Multiply \(x + 2\) by \(x^{3}\) to get \(x^{4}+2x^{3}\). Subtract this from the dividend: \((x^{4}-3x^{3}+0x^{2}+5x - 6)-(x^{4}+2x^{3})=-5x^{3}+0x^{2}+5x - 6\).

Step3: Divide the new leading term

Divide \(-5x^{3}\) by \(x\) to get \(-5x^{2}\). Multiply \(x + 2\) by \(-5x^{2}\) to get \(-5x^{3}-10x^{2}\). Subtract: \((-5x^{3}+0x^{2}+5x - 6)-(-5x^{3}-10x^{2}) = 10x^{2}+5x - 6\).

Step4: Divide the new leading term

Divide \(10x^{2}\) by \(x\) to get \(10x\). Multiply \(x + 2\) by \(10x\) to get \(10x^{2}+20x\). Subtract: \((10x^{2}+5x - 6)-(10x^{2}+20x)=-15x - 6\).

Step5: Divide the new leading term

Divide \(-15x\) by \(x\) to get \(-15\). Multiply \(x + 2\) by \(-15\) to get \(-15x-30\). Subtract: \((-15x - 6)-(-15x - 30)=24\).
So, \(\frac{x^{4}-3x^{3}+5x - 6}{x + 2}=x^{3}-5x^{2}+10x - 15+\frac{24}{x + 2}\) (if we consider the division with remainder) or if we made a mistake in the problem (maybe it's a typo and the original expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)\) instead of with the negative exponent? Wait, the original expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}\), but maybe there was a mis - writing. Wait, if we assume that the underlined part is a mistake and the expression is \((x^{4}-3x^{3}+5x - 6)\) divided by \((x + 2)^{-1}\), which is \((x^{4}-3x^{3}+5x - 6)(x + 2)\). Let's recalculate that.
If the expression is \((x^{4}-3x^{3}+5x - 6)(x + 2)\), then:
Multiply \(x^{4}\) by \(x\) and \(2\): \(x^{5}+2x^{4}\)
Multiply \(-3x^{3}\) by \(x\) and \(2\): \(-3x^{4}-6x^{3}\)
Multiply \(0x^{2}\) by \(x\) and \(2\): \(0x^{3}+0x^{2}\)
Multiply \(5x\) by \(x\) and \(2\): \(5x^{2}+10x\)
Multiply \(-6\) by \(x\) and \(2\): \(-6x-12\)
Now, combine like terms:
\(x^{5}+(2x^{4}-3x^{4})+(-6x^{3}+0x^{3})+(0x^{2}+5x^{2})+(10x - 6x)+(-12)\)
\(=x^{5}-x^{4}-6x^{3}+5x^{2}+4x - 12\)
Wait, the original problem has a fraction - like underlining, maybe the expression is \(\frac{(x^{4}-3x^{3}+5x - 6)(x + 2)^{-1}}{(x + 2)^{-1}}\). If that's the case,[SSE onError error]