Find the least common denominators of the rational expressions:3x2−12x2−x−6\frac{3x^2-12}{x^2-x-6}, 5x+10x2−4x+3\frac{5x+10}{x^2-4x+3}

(x+2)(x−3)(x−1)\left(x+2\right)\left(x-3\right)\left(x-1\right)(x+2)(x−3)(x−1)

Find the least common denominators of the rational expressions: 3 x 2 − 12 x 2 − x − 6 \frac{3x^2-12}{x^2-x-6} , 5 x + 10 x 2 − 4 x + 3 \frac{5x+10}{x^2-4x+3}

This problem asks us to find the least common denominators of the rational expressions, and we're now moving on to part b of this problem. Now to find the least common denominator, this is going to be a collection of all the unique prime factors that we can find multiplied all across. So the way that we can do this is by focusing on the denominators and factoring them completely. Now let's start with this denominator we have on the left side, where the numerator is three x squared minus 12. Notice this is a quadratic polynomial with one as the leading coefficient in front of x squared, so what I need to do is find a pair of numbers that multiply to give me negative six, and add to give me this negative one coefficient in front of the x. Now thinking about that pair of numbers, well, that's going to be two and negative three. Two times negative three gives me negative six. Two plus negative three gives me negative one. So what that's gonna give me is x plus two times x minus three. Now moving on to this next fraction that I see here, this next rational expression, we're gonna have five x plus 10 in the numerator, and then I need to factor the denominator. Notice that we have x squared minus 4x plus three. We can still see there's a one coefficient in front of the x squared, so we can just think about a pair of numbers that multiplies to give me three, but then adds to give me negative four. What pair of numbers would that be? Well, Well, if I think about it, that's just going to be negative one and negative three. These multiply to give me positive three, and they add to give me negative four. So this is going to break down into x minus one times x minus three. Now let's see what our least common denominator is going to be. Well, I can see we have an x plus two, but I don't see an x plus two anywhere else, nowhere in the denominators. So x plus two is a unique prime factor. And I see we have an x minus three, and I do see another x minus three that shows up there. We only need to write this once since we're looking for unique prime factors. And then I can see we have this x minus one, and this is the only x minus one that we have anywhere in these denominators, so we have x minus one. And this would be the LCD and the solution to part b of this problem.

Find the least common denominators of the rational expressions:3x2−12x2−x−6\frac{$$3x^2-12$$}{$$x^2-x-6$$}, 5x+10x2−4x+3\frac{5x+10}{$$x^2-4x+3$$}

(x+2)(x−3)(x−1)\left(x+2\right)\left(x-3\right)\left(x-1\right)(x+2)(x−3)(x−1)

(x+2)(x−3)(x+1)\left(x+2\right)\left(x-3\right)\left(x+1\right)(x+2)(x−3)(x+1)

(x−1)(x−3)\left(x-1\right)\left(x-3\right)(x−1)(x−3)

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

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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

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38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

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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

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38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

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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

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20m

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38m

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34m

7. Factoring2h 49m

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37m

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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

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27m

9. Roots, Radicals, and Complex Numbers2h 33m

Introduction to Radical Expressions
11m

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27m

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19m

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23m

Introduction to Complex Numbers
18m

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51m

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25m

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26m

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40m

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1h 28m

11. Inverse, Exponential, & Logarithmic Functions1h 5m

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25m

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13m

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26m

12. Conic Sections & Systems of Nonlinear Equations58m

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32m

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15m

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10m

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40m

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27m

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29m

Factorials
13m

8. Rational Expressions and Functions

Least Common Denominators

Intermediate Algebra

8. Rational Expressions and Functions

Least Common Denominators

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Multiple Choice

Find the least common denominators of the rational expressions:
$\frac{3 x^{2} - 12}{x^{2} - x - 6}$ , $\frac{5 x + 10}{x^{2} - 4 x + 3}$

$$\left$(x+2$\right$)$\left$(x-3$\right$)$\left$(x+1$\right$)$

$$\left$(x-1$\right$)$\left$(x-3$\right$)$

$$\left$(x+2$\right$)$\left$(x-3$\right$)$\left$(x-1$\right$)$

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

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Verified step by step guidance

Start by factoring the denominators of both rational expressions to identify their prime factors. For the first denominator $x^2 - x - 6$, find two numbers that multiply to $-6$ and add to $-1$.

Factor the first denominator as $\left(x - 3\right)\left(x + 2\right)$ based on the numbers found.

Next, factor the second denominator $x^2 - 4x + 3$ by finding two numbers that multiply to \$3$ and add to $-4$.

Factor the second denominator as $\left(x - 3\right)\left(x - 1\right)$ using those numbers.

To find the least common denominator (LCD), take each distinct factor from both denominators and multiply them together: $\left(x + 2\right)\left(x - 3\right)\left(x - 1\right)$.

6:33m

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Struggling with Intermediate Algebra?

Find the least common denominators of the rational expressions:3x2−12x2−x−6\(\frac{3x^2-12}{x^2-x-6}\), 5x+10x2−4x+3\(\frac{5x+10}{x^2-4x+3}\)

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Find the least common denominators of the rational expressions:
$\frac{3 x^{2} - 12}{x^{2} - x - 6}$ , $\frac{5 x + 10}{x^{2} - 4 x + 3}$