Let's divide the polynomial $2x^3 + 2x^2 + 5x - 5$ by $x^2 - x + 2$ using long division. Here's the process step by step:

Step 1: Set up the division

We will divide the dividend $2x^3 + 2x^2 + 5x - 5$ by the divisor $x^2 - x + 2$.

Step 2: Divide the leading terms

• Divide the leading term of the dividend $2x^3$ by the leading term of the divisor $x^2$. $\frac{2x^3}{x^2} = 2x$ This is the first term of the quotient.

Step 3: Multiply and subtract

• Multiply the entire divisor $x^2 - x + 2$ by $2x$: $2x(x^2 - x + 2) = 2x^3 - 2x^2 + 4x$
• Subtract this from the dividend: $(2x^3 + 2x^2 + 5x - 5) - (2x^3 - 2x^2 + 4x) = (2x^2 + 5x - 5) - (-2x^2 + 4x)$ Simplifying: $4x^2 + x - 5$

Step 4: Divide the leading terms again

• Divide the leading term of the new dividend $4x^2$ by the leading term of the divisor $x^2$: $\frac{4x^2}{x^2} = 4$ This is the next term of the quotient.

Step 5: Multiply and subtract again

• Multiply the divisor by $4$: $4(x^2 - x + 2) = 4x^2 - 4x + 8$
• Subtract this from the new dividend: $(4x^2 + x - 5) - (4x^2 - 4x + 8) = (4x^2 + x - 5) - (4x^2 - 4x + 8)$ Simplifying: $5x - 13$

Step 6: Write the final result

At this point, we cannot divide $5x - 13$ by $x^2 - x + 2$ since the degree of the remainder is less than the degree of the divisor. Thus, the remainder is $5x - 13$.

Final Answer:

The division is complete, and the result can be written in the form: $2x^3 + 2x^2 + 5x - 5 = (2x + 4)(x^2 - x + 2) + (5x - 13)$

Would you like further details or clarification on any step? Here are five related questions to consider:

1. How does polynomial long division compare to numerical long division?
2. What happens when the remainder is zero in polynomial division?
3. How can synthetic division be used as an alternative to long division for certain polynomials?
4. Why can't powers of $x$ be traded as they can with powers of 10 in decimal numbers?
5. How does the degree of the divisor affect the division