Find the least common denominators of the rational expressions:2y2+3y−5y2−4\frac{2y^2+3y-5}{y^2-4}, y−2y2+y−6\frac{y-2}{y^2+y-6}

(y+2)(y−2)(y+3)\left(y+2\right)\left(y-2\right)\left(y+3\right)(y+2)(y−2)(y+3)

Find the least common denominators of the rational expressions: 2 y 2 + 3 y − 5 y 2 − 4 \frac{2y^2+3y-5}{y^2-4} , y − 2 y 2 + y − 6 \frac{y-2}{y^2+y-6}

This problem wants us to find the least common denominators of the rational expressions. Now we're starting with part a of this problem, where we have these two rational expressions, and we want to find the least common denominator. Now the least common denominator is gonna be the product of all the unique terms that we can find. Now the best way to do this is to strictly factor these denominators here to see what terms or factors are unique. Now starting with this first fraction where we have two y squared plus three y minus five on top, notice that we have y squared minus four. Y squared minus four is the same thing as y squared minus two squared. And this is the difference between two squares. So y squared minus two squared can be factored into y plus two times y minus two. And that's what happens when we fully factor the denominator for this first fraction that we see. Now for the second fraction where we have, or I should say rational expression, where we have y minus two on top, in the denominator we're going to have y squared plus y minus six. What I can do is think about a pair of numbers that multiply to give me negative six and add to give me this one coefficient in front of the y. Because that's what I'm looking for since I can see we have a one coefficient in front of the y squared, I can just factor this like a typical quadratic polynomial. And And if I think about that pair of numbers, well, that's going to be negative two and three. Because negative two times three gives me negative six. Negative two plus three gives me positive one in front of this y. So that means this is going to be factored into x minus or y minus two, I should say, y minus two times y plus three. So we now have the denominators fully factored, and I need to look for unique terms or unique factors in the denominator. Now you can see we have a y plus two right here, and I don't see a y plus two anywhere else. So y plus two is unique. Now you can see we have a y minus two, but we also have a y minus two there. So I'm only gonna write that y minus two once. I don't need to write it twice. And then I can see we have a y plus three. That's the only y plus three that we have. So this means that y plus three will be this last term, and multiplying those all across will give me the LCD as a solution to part a of this problem.

Find the least common denominators of the rational expressions:2y2+3y−5y2−4\frac{$$2y^2+3y-5$$}{$$y^2-4$$}, y−2y2+y−6\frac{y-2}{$$y^2+y-6$$}

(y+2)(y−2)(y+3)\left(y+2\right)\left(y-2\right)\left(y+3\right)(y+2)(y−2)(y+3)

(y−2)(y−2)(y+3)\left(y-2\right)\left(y-2\right)\left(y+3\right)(y−2)(y−2)(y+3)

Table of contents

Skip topic navigation

1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers2h 33m

Introduction to Radical Expressions
11m

Simplifying Radical Expressions
27m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
23m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions1h 5m

Introduction to Inverse Functions
25m

Exponential Functions
13m

Introduction to Logarithmic Functions
26m

12. Conic Sections & Systems of Nonlinear Equations58m

Graphing Circles
32m

Ellipses
15m

Parabolas
10m

13. Sequences, Series, and the Binomial Theorem1h 51m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

8. Rational Expressions and Functions

Least Common Denominators

Intermediate Algebra

8. Rational Expressions and Functions

Least Common Denominators

Previous problemNext problem

Struggling with Intermediate Algebra?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Find the least common denominators of the rational expressions:
$\frac{2 y^{2} + 3 y - 5}{y^{2} - 4}$ , $\frac{y - 2}{y^{2} + y - 6}$

$$\left$(y-2$\right$)$\left$(y-2$\right$)$\left$(y+3$\right$)$

$y^2-4$

$y^2+y-6$

$$\left$(y+2$\right$)$\left$(y-2$\right$)$\left$(y+3$\right$)$

0 Comments

Previous problemNext problem

Verified step by step guidance

Identify the denominators of the given rational expressions: the first denominator is $y^2 - 4$ and the second denominator is $y^2 + y - 6$.

Factor each denominator completely using factoring techniques: recognize $y^2 - 4$ as a difference of squares and factor it as $(y - 2)(y + 2)$.

Factor the second denominator $y^2 + y - 6$ by finding two numbers that multiply to $-6$ and add to \$1$, resulting in $(y + 3)(y - 2)$.

Determine the least common denominator (LCD) by taking each distinct factor from both denominators at its highest power: include $(y + 2)$, $(y - 2)$, and $(y + 3)$.

Write the LCD as the product of these factors: $\left(y+2\right)\left(y-2\right)\left(y+3\right)$.

6:33m

Watch next

Master Least Common Denominators of Rational Expressions with a bite sized video explanation from Patrick Ford

Struggling with Intermediate Algebra?

Find the least common denominators of the rational expressions:2y2+3y−5y2−4\(\frac{2y^2+3y-5}{y^2-4}\), y−2y2+y−6\(\frac{y-2}{y^2+y-6}\)

Watch next

Find the least common denominators of the rational expressions:
$\frac{2 y^{2} + 3 y - 5}{y^{2} - 4}$ , $\frac{y - 2}{y^{2} + y - 6}$