Math Problem Statement

(2x*2-x+4)/(x+3)

Solution

To simplify the expression

2x2x+4x+3\frac{2x^2 - x + 4}{x + 3}

we can attempt polynomial division to see if it can be broken down further.

Step 1: Polynomial Division

  1. Divide the leading term in the numerator, 2x22x^2, by the leading term in the denominator, xx, which gives 2x2x.

  2. Multiply 2x2x by x+3x + 3, resulting in 2x2+6x2x^2 + 6x.

  3. Subtract 2x2+6x2x^2 + 6x from 2x2x+42x^2 - x + 4, giving 7x+4-7x + 4.

  4. Now, repeat the process with 7x-7x by dividing by xx, resulting in 7-7.

  5. Multiply 7-7 by x+3x + 3, resulting in 7x21-7x - 21.

  6. Subtract 7x21-7x - 21 from 7x+4-7x + 4, which leaves a remainder of 2525.

Final Result

Thus, we have:

2x2x+4x+3=2x7+25x+3\frac{2x^2 - x + 4}{x + 3} = 2x - 7 + \frac{25}{x + 3}

So, the simplified form of the expression is:

2x7+25x+32x - 7 + \frac{25}{x + 3}

Would you like further details on any steps, or do you have any additional questions?


Here are some related questions that might interest you:

  1. How would this division change if the numerator were 2x2x+52x^2 - x + 5?
  2. What are some common mistakes to avoid in polynomial division?
  3. How does polynomial division differ from synthetic division?
  4. Could this expression be further simplified if additional terms were added?
  5. How can you confirm the correctness of polynomial division?

Tip: Always double-check each step in polynomial division, especially when dealing with negative terms, to avoid errors in signs.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Division
Rational Expressions

Formulas

Polynomial Division Process

Theorems

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Suitable Grade Level

Grades 9-11