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Questions Math Without Calculator Question 11

Reviewed by Calculator Editorial Team

Question 11 from a typical math exam without calculator requires solving a complex algebraic expression. This guide provides a complete solution method, step-by-step instructions, and a calculator to verify your work.

Understanding the Problem

The problem typically presents a complex algebraic expression that must be simplified without using a calculator. Understanding the structure of the expression is crucial for solving it correctly.

Example Problem

Simplify the expression: (3x² - 5x + 2) / (x - 1)

To solve this problem, you'll need to perform polynomial division. The goal is to rewrite the expression in a simpler form that reveals its roots and behavior.

Step-by-Step Solution

Follow these steps to simplify the expression:

  1. Identify the dividend and divisor: The dividend is 3x² - 5x + 2, and the divisor is x - 1.
  2. Perform polynomial long division:
    1. Divide the first term of the dividend (3x²) by the first term of the divisor (x) to get 3x.
    2. Multiply the entire divisor (x - 1) by 3x to get 3x² - 3x.
    3. Subtract this from the dividend to get -2x + 2.
    4. Divide the first term of the new polynomial (-2x) by the first term of the divisor (x) to get -2.
    5. Multiply the entire divisor by -2 to get -2x + 2.
    6. Subtract this from the previous result to get 0.
  3. Write the final simplified form: The simplified form is 3x - 2.

Formula Used

Polynomial long division is used to simplify the expression (3x² - 5x + 2) / (x - 1).

Common Mistakes to Avoid

When solving this problem, avoid these common errors:

  • Incorrect division steps: Ensure each division step is performed accurately.
  • Sign errors: Pay attention to positive and negative signs during subtraction.
  • Skipping terms: Do not skip any terms in the polynomial division process.

Double-check each step to ensure accuracy. Polynomial division requires careful attention to detail.

Verification of the Solution

To verify your solution, multiply the simplified form by the divisor and check if you get back the original dividend.

Verification Example

(3x - 2)(x - 1) = 3x² - 3x - 2x + 2 = 3x² - 5x + 2

This confirms that the simplified form is correct.

Frequently Asked Questions

What is polynomial long division?
Polynomial long division is a method for dividing one polynomial by another, similar to numerical long division.
When should I use polynomial long division?
Use polynomial long division when you need to simplify complex algebraic expressions without a calculator.
How do I know if I've done it correctly?
Multiply the simplified form by the divisor to check if you get back the original dividend.
What if I make a mistake during division?
Review each step carefully and redo any incorrect calculations.
Can I use this method for any polynomial?
Yes, polynomial long division can be used for any polynomial as long as the degree of the divisor is less than or equal to the degree of the dividend.