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Find the sum or difference. Simplify. 7x² –x-4 6x²+12x-44 1. x²-25 x²-25 3x 4 2. x2+2x-8 x2+3x-10 2n 13n+15 n+5 n-3 n²+2n-15 3. 3.

Find the sum or difference. Simplify. 7x² –x-4 6x²+12x-44 1. x²-25 x²-25 3x 4 2. x2+2x-8 x2+3x-10 2n 13n+15 n+5 n-3 n²+2n-15 3. 3.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Topic: Simplifying Rational Expressions** **Find the sum or difference. Simplify.** 1. \[\frac{7x^2 - x - 4}{x^2 - 25} - \frac{6x^2 + 12x - 44}{x^2 - 25}\] 2. \[\frac{3x}{x^2 + 2x - 8} + \frac{4}{x^2 + 3x - 10}\] 3. \[\frac{2n}{n + 5} + \frac{3}{n - 3} - \frac{13n + 15}{n^2 + 2n - 15}\] --- **Find each product or quotient. Simplify.** 4. \[\frac{2y^2 - 3y - 14}{25 - y^2} \cdot \frac{y^2 - 10y + 25}{2y^2 - 13y + 21}\] --- **Instructions:** - Identify the common denominator for each expression. - Simplify by factoring where possible. - Combine terms and simplify the final expression.
Transcribed Image Text:**Topic: Simplifying Rational Expressions** **Find the sum or difference. Simplify.** 1. \[\frac{7x^2 - x - 4}{x^2 - 25} - \frac{6x^2 + 12x - 44}{x^2 - 25}\] 2. \[\frac{3x}{x^2 + 2x - 8} + \frac{4}{x^2 + 3x - 10}\] 3. \[\frac{2n}{n + 5} + \frac{3}{n - 3} - \frac{13n + 15}{n^2 + 2n - 15}\] --- **Find each product or quotient. Simplify.** 4. \[\frac{2y^2 - 3y - 14}{25 - y^2} \cdot \frac{y^2 - 10y + 25}{2y^2 - 13y + 21}\] --- **Instructions:** - Identify the common denominator for each expression. - Simplify by factoring where possible. - Combine terms and simplify the final expression.
**Educational Website Content: Simplifying Rational Expressions** For exercises involving the simplification of rational expressions, follow the examples below. Each step employs algebraic techniques to reduce the expressions to their simplest form. --- **Exercise 5:** Simplify the expression \[ \frac{4y^2 + 7y - 2}{35y + 10} \div \frac{y^2 - 4}{7y^2 - 12y - 4} \] **Exercise 6:** Simplify the expression \[ \frac{4a^2 + 32a}{3a + 2} \cdot \frac{3a^2 - a - 2}{a^2 + a - 30} \div \frac{108a^2 - 24a}{a + 6} \] **Instructions:** Simplify each rational expression completely. **Exercise 7:** Simplify: \[ \frac{\frac{x - 2x}{2}}{x - 4 + 2} + \frac{2}{x + 4} \] **Exercise 8:** Simplify: \[ \frac{\frac{3}{x + 6} + \frac{5}{x - 6}}{\frac{1}{x^2 - 36} + \frac{x}{x + 6}} \] **Exercise 9:** Simplify: \[ \frac{\frac{y}{y + 3}}{2 + \frac{1}{y - 3}} \] --- Each of these problems requires you to apply the concepts of factoring, finding common denominators, and performing arithmetic operations on fractions to simplify the expressions fully. As you work through these exercises, remember to check for common factors and ensure all parts of the expression are simplified correctly.
Transcribed Image Text:**Educational Website Content: Simplifying Rational Expressions** For exercises involving the simplification of rational expressions, follow the examples below. Each step employs algebraic techniques to reduce the expressions to their simplest form. --- **Exercise 5:** Simplify the expression \[ \frac{4y^2 + 7y - 2}{35y + 10} \div \frac{y^2 - 4}{7y^2 - 12y - 4} \] **Exercise 6:** Simplify the expression \[ \frac{4a^2 + 32a}{3a + 2} \cdot \frac{3a^2 - a - 2}{a^2 + a - 30} \div \frac{108a^2 - 24a}{a + 6} \] **Instructions:** Simplify each rational expression completely. **Exercise 7:** Simplify: \[ \frac{\frac{x - 2x}{2}}{x - 4 + 2} + \frac{2}{x + 4} \] **Exercise 8:** Simplify: \[ \frac{\frac{3}{x + 6} + \frac{5}{x - 6}}{\frac{1}{x^2 - 36} + \frac{x}{x + 6}} \] **Exercise 9:** Simplify: \[ \frac{\frac{y}{y + 3}}{2 + \frac{1}{y - 3}} \] --- Each of these problems requires you to apply the concepts of factoring, finding common denominators, and performing arithmetic operations on fractions to simplify the expressions fully. As you work through these exercises, remember to check for common factors and ensure all parts of the expression are simplified correctly.
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\frac{7x^2-x-4}{x^2-25}-\frac{6x^2+12x-44}{x^2-25}

=\frac{7x^2-x-4-\left(6x^2+12x-44\right)}{x^2-25}

=\frac{7x^2-x-4-6x^2-12x+44}{x^2-25}

=\frac{x^2-13x+40}{x^2-25}

=\frac{\left(x^2-5x\right)+\left(-8x+40\right)}{x^2-25}

=\frac{ x\left(x-5\right)-8\left(x-5\right)}{x^2-25}

=\frac{\left(x-5\right)\left(x-8\right)}{x^2-25}

=\frac{\left(x-5\right)\left(x-8\right)}{\left(x+5\right)\left(x-5\right)}

{\color{Red} =\frac{x-8}{x+5}}

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