Rational Functions of Expressions are ratios (or quotients) of polynomials. They are processed exactly like fractions. Polynomials in standard form are sums of powers of the variable. Numbers in decimal representation are sums of powers of 10. Polynomials are processed much like numbers, except that you cannot trade some powers of 𝑥 for a higher power of 𝑥. Thus the manipulation of polynomials is in some ways actually easier than the corresponding manipulation of decimal numbers! An example for this fact is provided by long division.
Use long division to divide the polynomial these two polynomials then write your answer in the form "dividend = quotient × divisor + remainder".
2𝑥3+2𝑥2+5𝑥−5=(𝑥2−𝑥+2)

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

Use long division to divide the polynomials 2x^3 + 2x^2 + 5x - 5 by x^2 - x + 2, and write your answer in the form 'dividend = quotient × divisor + remainder'.

Learn how to divide the polynomial 2x^3 + 2x^2 + 5x - 5 by x^2 - x + 2 using long division. This solution breaks down the process step by step, showing how to find the quotient and remainder. Ideal for students in Grades 9-12.

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