Add And State The Sum In Simplest Form.1. $\frac{2x+3}{x+4}+\frac{x-1}{x^2+2x-8}$2. $\frac{2x^2-7}{x^2+2x-8}$3. $\frac{2x^2+x-7}{x^2+2x-8}$4. $\frac{3x+2}{x^2+2x-8}$5. $\frac{3x+2}{x^2+3x-4}$

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**Simplifying Algebraic Expressions: A Step-by-Step Guide**

What are Algebraic Expressions?

Algebraic expressions are a combination of variables, constants, and mathematical operations. They are used to represent a value or a relationship between values. In this article, we will focus on simplifying algebraic expressions, which is an essential skill in mathematics.

Why Simplify Algebraic Expressions?

Simplifying algebraic expressions is crucial in mathematics because it helps to:

  • Reduce the complexity of an expression
  • Make it easier to solve equations and inequalities
  • Identify patterns and relationships between variables
  • Improve the accuracy of calculations

How to Simplify Algebraic Expressions

Simplifying algebraic expressions involves several steps:

  1. Combine like terms: Combine terms that have the same variable and exponent.
  2. Simplify fractions: Simplify fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
  3. Cancel out common factors: Cancel out common factors between the numerator and denominator.
  4. Use the distributive property: Use the distributive property to expand and simplify expressions.

Example 1: Simplifying a Fraction

Suppose we want to simplify the fraction 2x+3x+4+xβˆ’1x2+2xβˆ’8\frac{2x+3}{x+4}+\frac{x-1}{x^2+2x-8}.

Step 1: Factor the denominator

The denominator x2+2xβˆ’8x^2+2x-8 can be factored as (x+4)(xβˆ’2)(x+4)(x-2).

Step 2: Simplify the fractions

The fraction 2x+3x+4\frac{2x+3}{x+4} can be simplified as 2x+3x+4=2(x+4)βˆ’7x+4=2βˆ’7x+4\frac{2x+3}{x+4} = \frac{2(x+4)-7}{x+4} = 2 - \frac{7}{x+4}.

The fraction xβˆ’1x2+2xβˆ’8\frac{x-1}{x^2+2x-8} can be simplified as xβˆ’1x2+2xβˆ’8=xβˆ’1(x+4)(xβˆ’2)=1xβˆ’2βˆ’1x+4\frac{x-1}{x^2+2x-8} = \frac{x-1}{(x+4)(x-2)} = \frac{1}{x-2} - \frac{1}{x+4}.

Step 3: Combine like terms

The simplified expression is 2βˆ’7x+4+1xβˆ’2βˆ’1x+42 - \frac{7}{x+4} + \frac{1}{x-2} - \frac{1}{x+4}.

Step 4: Cancel out common factors

The common factor βˆ’1x+4-\frac{1}{x+4} can be canceled out.

Step 5: Simplify the expression

The simplified expression is 2+1xβˆ’22 + \frac{1}{x-2}.

Example 2: Simplifying a Rational Expression

Suppose we want to simplify the rational expression 2x2βˆ’7x2+2xβˆ’8\frac{2x^2-7}{x^2+2x-8}.

Step 1: Factor the denominator

The denominator x2+2xβˆ’8x^2+2x-8 can be factored as (x+4)(xβˆ’2)(x+4)(x-2).

Step 2: Simplify the numerator

The numerator 2x2βˆ’72x^2-7 can be factored as 2(x2βˆ’72)2(x^2- \frac{7}{2}).

Step 3: Simplify the fraction

The fraction 2x2βˆ’7x2+2xβˆ’8\frac{2x^2-7}{x^2+2x-8} can be simplified as 2(x2βˆ’72)(x+4)(xβˆ’2)=2(xβˆ’2)(x+4)βˆ’7(x+4)(x+4)(xβˆ’2)=2βˆ’7x+4\frac{2(x^2- \frac{7}{2})}{(x+4)(x-2)} = \frac{2(x-2)(x+4)-7(x+4)}{(x+4)(x-2)} = 2 - \frac{7}{x+4}.

Step 4: Simplify the expression

The simplified expression is 2βˆ’7x+42 - \frac{7}{x+4}.

Example 3: Simplifying a Rational Expression with a Common Factor

Suppose we want to simplify the rational expression 2x2+xβˆ’7x2+2xβˆ’8\frac{2x^2+x-7}{x^2+2x-8}.

Step 1: Factor the denominator

The denominator x2+2xβˆ’8x^2+2x-8 can be factored as (x+4)(xβˆ’2)(x+4)(x-2).

Step 2: Simplify the numerator

The numerator 2x2+xβˆ’72x^2+x-7 can be factored as (x+4)(2xβˆ’3)(x+4)(2x-3).

Step 3: Simplify the fraction

The fraction 2x2+xβˆ’7x2+2xβˆ’8\frac{2x^2+x-7}{x^2+2x-8} can be simplified as (x+4)(2xβˆ’3)(x+4)(xβˆ’2)=2xβˆ’3xβˆ’2\frac{(x+4)(2x-3)}{(x+4)(x-2)} = \frac{2x-3}{x-2}.

Step 4: Simplify the expression

The simplified expression is 2xβˆ’3xβˆ’2\frac{2x-3}{x-2}.

Example 4: Simplifying a Rational Expression with a Common Factor

Suppose we want to simplify the rational expression 3x+2x2+2xβˆ’8\frac{3x+2}{x^2+2x-8}.

Step 1: Factor the denominator

The denominator x2+2xβˆ’8x^2+2x-8 can be factored as (x+4)(xβˆ’2)(x+4)(x-2).

Step 2: Simplify the numerator

The numerator 3x+23x+2 can be factored as (3x+2)(3x+2).

Step 3: Simplify the fraction

The fraction 3x+2x2+2xβˆ’8\frac{3x+2}{x^2+2x-8} can be simplified as 3x+2(x+4)(xβˆ’2)=3(x+4)βˆ’14(x+4)(xβˆ’2)=3βˆ’14x+4\frac{3x+2}{(x+4)(x-2)} = \frac{3(x+4)-14}{(x+4)(x-2)} = 3 - \frac{14}{x+4}.

Step 4: Simplify the expression

The simplified expression is 3βˆ’14x+43 - \frac{14}{x+4}.

Example 5: Simplifying a Rational Expression with a Common Factor

Suppose we want to simplify the rational expression 3x+2x2+3xβˆ’4\frac{3x+2}{x^2+3x-4}.

Step 1: Factor the denominator

The denominator x2+3xβˆ’4x^2+3x-4 can be factored as (x+4)(xβˆ’1)(x+4)(x-1).

Step 2: Simplify the numerator

The numerator 3x+23x+2 can be factored as (3x+2)(3x+2).

Step 3: Simplify the fraction

The fraction 3x+2x2+3xβˆ’4\frac{3x+2}{x^2+3x-4} can be simplified as 3x+2(x+4)(xβˆ’1)=3(x+4)βˆ’14(x+4)(xβˆ’1)=3βˆ’14x+4\frac{3x+2}{(x+4)(x-1)} = \frac{3(x+4)-14}{(x+4)(x-1)} = 3 - \frac{14}{x+4}.

Step 4: Simplify the expression

The simplified expression is 3βˆ’14x+43 - \frac{14}{x+4}.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can simplify rational expressions and make them easier to solve. Remember to combine like terms, simplify fractions, cancel out common factors, and use the distributive property to expand and simplify expressions. With practice, you will become proficient in simplifying algebraic expressions and be able to solve equations and inequalities with ease.

Frequently Asked Questions

  • Q: What is the difference between a rational expression and a rational number? A: A rational number is a number that can be expressed as the ratio of two integers, while a rational expression is an algebraic expression that can be expressed as the ratio of two polynomials.
  • Q: How do I simplify a rational expression? A: To simplify a rational expression, you need to combine like terms, simplify fractions, cancel out common factors, and use the distributive property to expand and simplify expressions.
  • Q: What is the greatest common divisor (GCD) of two numbers? A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.
  • 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 use the distributive property to expand and simplify the expression.
  • Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.