Question

1. \\((x^2 - 10x - 24) \div (x + 2)\\)\
2. \\((b^4 - 2b^3 + b^2 - 3b + 4)(b - 2)^{-1}\\)\
3. \\((x^5 - 3x^2 - 20) \div (x - 2)\\)

Explanation:

Problem 3:

Step 1: Factor the numerator

We factor \(x^2 - 10x - 24\). We need two numbers that multiply to \(-24\) and add to \(-10\). The numbers are \(-12\) and \(2\). So, \(x^2 - 10x - 24=(x - 12)(x + 2)\)

Step 2: Divide by the denominator

Now we have \(\frac{(x - 12)(x + 2)}{x + 2}\). We can cancel out the common factor \((x + 2)\) (assuming \(x
eq - 2\))

Step 1: Rewrite the expression

The expression \((b^4 - 2b^3 + b^2 - 3b + 4)(b - 2)^{-1}\) is the same as \(\frac{b^4 - 2b^3 + b^2 - 3b + 4}{b - 2}\)

Step 2: Perform polynomial long division

Divide \(b^4 - 2b^3 + b^2 - 3b + 4\) by \(b - 2\)

• Divide \(b^4\) by \(b\) to get \(b^3\). Multiply \((b - 2)\) by \(b^3\) to get \(b^4-2b^3\). Subtract this from the dividend: \((b^4 - 2b^3 + b^2 - 3b + 4)-(b^4 - 2b^3)=b^2 - 3b + 4\)
• Divide \(b^2\) by \(b\) to get \(b\). Multiply \((b - 2)\) by \(b\) to get \(b^2-2b\). Subtract this from \(b^2 - 3b + 4\): \((b^2 - 3b + 4)-(b^2 - 2b)=-b + 4\)
• Divide \(-b\) by \(b\) to get \(-1\). Multiply \((b - 2)\) by \(-1\) to get \(-b + 2\). Subtract this from \(-b + 4\): \((-b + 4)-(-b + 2)=2\)

So, \(\frac{b^4 - 2b^3 + b^2 - 3b + 4}{b - 2}=b^3 + b - 1+\frac{2}{b - 2}\)

Step 1: Perform polynomial long division

Divide \(x^5-3x^2 - 20\) by \(x - 2\)

• Divide \(x^5\) by \(x\) to get \(x^4\). Multiply \((x - 2)\) by \(x^4\) to get \(x^5-2x^4\). Subtract this from the dividend: \((x^5-3x^2 - 20)-(x^5 - 2x^4)=2x^4-3x^2 - 20\)
• Divide \(2x^4\) by \(x\) to get \(2x^3\). Multiply \((x - 2)\) by \(2x^3\) to get \(2x^4-4x^3\). Subtract this from \(2x^4-3x^2 - 20\): \((2x^4-3x^2 - 20)-(2x^4 - 4x^3)=4x^3-3x^2 - 20\)
• Divide \(4x^3\) by \(x\) to get \(4x^2\). Multiply \((x - 2)\) by \(4x^2\) to get \(4x^3-8x^2\). Subtract this from \(4x^3-3x^2 - 20\): \((4x^3-3x^2 - 20)-(4x^3 - 8x^2)=5x^2 - 20\)
• Divide \(5x^2\) by \(x\) to get \(5x\). Multiply \((x - 2)\) by \(5x\) to get \(5x^2-10x\). Subtract this from \(5x^2 - 20\): \((5x^2 - 20)-(5x^2 - 10x)=10x - 20\)
• Divide \(10x\) by \(x\) to get \(10\). Multiply \((x - 2)\) by \(10\) to get \(10x-20\). Subtract this from \(10x - 20\): \((10x - 20)-(10x - 20)=0\)

Answer:

\(x - 12\)

Problem 4: