6. Systems of Equations and Inequalities

Decompose P / Q, Where Q Has Repeated Linear Factors

Partial Fractions

Partial fractions are SPECIAL! (Repeated linear factors)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)

In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x^2 +2x -3)

Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: 3/(x−4) − 2/(x+2).

Partial Fraction Decomposition - Example 4

Partial Fraction Decomposition

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x^3-4x^2+9x-5)/(x^2 -2x+3)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (7x-4)/(x^2-x-12)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression.5x^2+6x+3/(x + 1)(x² + 2x + 2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4x^2+13x-9/x (x − 1)(x+3)

Partial fractions with an irreducible quadratic factor

Partial Fractions Repeated Quadratic Factors

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^3 + 5x^2 + 7x - 1)/(x^2 + x + 1)^2

Repeated Quadratic Factors

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum: <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/54bb9cb079e5a7bd-1678180384827.jpg" style="width: 300px;">

Partial Fraction Decomposition - Example 1

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x²+2x+7/x(x − 1)^2

Partial Fractions Repeated Linear Factors

Partial Fraction Decomposition - Example 2

Pre-Calculus - Partial Fraction Decomposition

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x^3 + x² /(x² + 4)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

Ex 6: Partial Fraction Decomposition (Repeating Quadratic Factors)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^2/(x − 1)² (x + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (6x-11)/(x − 1)²

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.(11x - 10)/(x − 2) (x + 1)

PC partial fraction decomposition (with repeated factors)

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (6x^2-14x-27)/(x+2) (x − 3)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

Partial Fractions Quadratic Factors

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4/(2x^2 -5x -3)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (2x^2 -18x -12)/x³- 4x

Partial fractions, repeated quadratic factors

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (7x^2 -9x+3)/(x²+7)²

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 6x^2-x+1/(x^3 + x²+x+1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 5x^2 -6x+7/(x − 1) (x² + 1)

Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x^2+1) + 2x/(x^2+1)^2.

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 5x²-6x+7 /(x − 1) (x² + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4x² - 7x - 3/(x^3 -x)

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^5+2)/(x^2-1)

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^4-x^2+2)/(x^3-x^2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^3+x^2+2/(x² + 2)²

Partial Fraction Decomposition - Example 6

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (3x+16)/(x + 1) (x − 2)²

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (4x^2+3x+14)/(x^3 - 8)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x^2-6x+3)/(x − 2)³

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x-2)(x-3)

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College Algebra

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Graph a Linear Inequality in Two Variables

Graph a Nonlinear Inequality in Two Variables

Use Mathematical Models Involving Linear Inequalities

Graph a System of Inequalities

Systems of Inequalities

Determine Whether an Ordered Pair is a Solution of a Linear System

Solve Linear Systems by Substitution

Solve Linear Systems by Addition

Identify Systems That Don't Have One Ordered-Pair Solution