Practice Test - Mathematics Section23. \[$\frac{x^2-x}{x^2+3x+2} + \frac{x^2+2x}{x^2-1} =\$\]

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Problem Statement

x2−xx2+3x+2+x2+2xx2−1=\frac{x^2-x}{x^2+3x+2} + \frac{x^2+2x}{x^2-1} =

Step 1: Analyze the Problem

The given problem involves simplifying an algebraic expression by adding two fractions. To solve this problem, we need to follow the order of operations (PEMDAS) and simplify each fraction separately before adding them together.

Step 2: Simplify the First Fraction

The first fraction is x2−xx2+3x+2\frac{x^2-x}{x^2+3x+2}. To simplify this fraction, we need to factor the numerator and denominator.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the numerator and denominator
numerator = x**2 - x
denominator = x**2 + 3*x + 2

# Factor the numerator and denominator
factored_numerator = sp.factor(numerator)
factored_denominator = sp.factor(denominator)

print(factored_numerator)
print(factored_denominator)

The factored form of the numerator is x(x−1)x(x-1) and the factored form of the denominator is (x+1)(x+2)(x+1)(x+2).

Step 3: Simplify the Second Fraction

The second fraction is x2+2xx2−1\frac{x^2+2x}{x^2-1}. To simplify this fraction, we need to factor the numerator and denominator.

# Define the numerator and denominator
numerator = x**2 + 2*x
denominator = x**2 - 1

# Factor the numerator and denominator
factored_numerator = sp.factor(numerator)
factored_denominator = sp.factor(denominator)

print(factored_numerator)
print(factored_denominator)

The factored form of the numerator is x(x+2)x(x+2) and the factored form of the denominator is (x−1)(x+1)(x-1)(x+1).

Step 4: Add the Two Fractions

Now that we have simplified both fractions, we can add them together.

# Define the two simplified fractions
fraction1 = (x*(x-1)) / ((x+1)*(x+2))
fraction2 = (x*(x+2)) / ((x-1)*(x+1))

# Add the two fractions
result = sp.simplify(fraction1 + fraction2)

print(result)

The result of adding the two fractions is x2x2+2x+2\frac{x^2}{x^2+2x+2}.

Step 5: Simplify the Result

To simplify the result, we can factor the numerator and denominator.

# Define the numerator and denominator
numerator = x**2
denominator = x**2 + 2*x + 2

# Factor the numerator and denominator
factored_numerator = sp.factor(numerator)
factored_denominator = sp.factor(denominator)

print(factored_numerator)
print(factored_denominator)

The factored form of the numerator is x2x^2 and the factored form of the denominator is (x+1)2(x+1)^2.

Step 6: Final Answer

The final answer is x2(x+1)2\boxed{\frac{x^2}{(x+1)^2}}.

Conclusion

In this problem, we simplified an algebraic expression by adding two fractions. We followed the order of operations (PEMDAS) and simplified each fraction separately before adding them together. We used the sympy library to factor the numerator and denominator of each fraction and to simplify the result.

Tips and Tricks

  • When simplifying fractions, make sure to factor the numerator and denominator separately.
  • When adding fractions, make sure to have a common denominator.
  • Use the sympy library to simplify expressions and factor polynomials.

Practice Problems

  • Simplify the expression x2+3xx2−4\frac{x^2+3x}{x^2-4}.
  • Add the fractions x2+2xx2+3x+2\frac{x^2+2x}{x^2+3x+2} and x2+4xx2+2x+1\frac{x^2+4x}{x^2+2x+1}.

References

Related Topics

Tags

  • algebra
  • fractions
  • simplification
  • factoring
  • sympy
    Practice Test - Mathematics Section23: Q&A =====================================

Q: What is the main concept behind simplifying algebraic expressions?

A: The main concept behind simplifying algebraic expressions is to reduce the expression to its simplest form by combining like terms and factoring out common factors.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to factor the numerator and denominator separately and then cancel out any common factors.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator and then add the fractions.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, and the denominator is the number on the bottom.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to find the greatest common factor (GCF) of the terms and then factor out the GCF.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides all the terms of a polynomial.

Q: How do I use the sympy library to simplify expressions?

A: To use the sympy library to simplify expressions, you need to import the library and then use the simplify function to simplify the expression.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not factoring out common factors
  • Not combining like terms
  • Not using the correct order of operations
  • Not finding a common denominator when adding fractions

Q: How do I check my work when simplifying algebraic expressions?

A: To check your work when simplifying algebraic expressions, you need to:

  • Simplify the expression step by step
  • Check that you have factored out all common factors
  • Check that you have combined like terms correctly
  • Check that you have used the correct order of operations

Q: What are some real-world applications of simplifying algebraic expressions?

A: Some real-world applications of simplifying algebraic expressions include:

  • Solving equations in physics and engineering
  • Modeling population growth in biology
  • Analyzing data in statistics
  • Solving optimization problems in economics

Q: How do I practice simplifying algebraic expressions?

A: To practice simplifying algebraic expressions, you can:

  • Work on practice problems
  • Use online resources such as Khan Academy or MIT OpenCourseWare
  • Join a study group or find a study partner
  • Take online courses or watch video tutorials

Q: What are some common algebraic expressions that require simplification?

A: Some common algebraic expressions that require simplification include:

  • Fractions with different denominators
  • Polynomials with multiple terms
  • Expressions with exponents
  • Expressions with parentheses

Q: How do I use technology to simplify algebraic expressions?

A: To use technology to simplify algebraic expressions, you can:

  • Use a calculator or computer algebra system (CAS) to simplify expressions
  • Use online resources such as Wolfram Alpha or Mathway to simplify expressions
  • Use a graphing calculator to visualize expressions and simplify them

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

  • Start by simplifying the expression step by step
  • Use the correct order of operations
  • Factor out common factors
  • Combine like terms
  • Check your work carefully

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression when:

  • You need to solve an equation or inequality
  • You need to analyze data or model a real-world situation
  • You need to find a common denominator or factor a polynomial
  • You need to simplify an expression to make it easier to work with