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)

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

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

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

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1. Review of Real Numbers2h 24m

Simplifying Fractions
30m

Evaluating Exponents
14m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

2. Linear Equations and Inequalities3h 42m

The Addition and Subtraction Properties of Equality
47m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Solving Linear Inequalities
1h 8m

3. Solving Word Problems2h 48m

Introduction to Problem Solving
37m

Formulas
21m

Percent Problem Solving
1h 4m

Mixture Problem Solving
43m

4. Graphing4h 42m

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
40m

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15m

5. Systems of Linear Equations2h 6m

Solving Systems of Linear Equations by Graphing
41m

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15m

Solving Systems of Linear Equations by Elimination
45m

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23m

6. Exponents and Polynomials3h 25m

The Product Rule
10m

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13m

Intro to the Power Rules
18m

The Power of a Quotient Rule
18m

Negative Exponents
28m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

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20m

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33m

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34m

7. Factoring2h 36m

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37m

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11m

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

Factoring Special Products
33m

Quadratic Equations & Applications
35m

8. Rational Expressions and Equations3h 51m

Simplifying Rational Expressions
39m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
24m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
39m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots and Radicals2h 46m

Introduction to Radical Expressions
47m

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46m

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19m

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53m

10. Quadratic Equations3h 2m

The Square Root Property
17m

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24m

The Quadratic Formula
29m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

Complex Solutions of Quadratic Equations
10m

Graphing Quadratic Equations
30m

8. Rational Expressions and Equations

Least Common Denominators

Beginning Algebra

8. Rational Expressions and Equations

Least Common Denominators

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Multiple Choice

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

$(x + 2) (x - 3) (x + 1)$

$(x - 1) (x - 3)$

$(x + 2) (x - 3) (x - 1)$

$(x - 2) (x + 3)$

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Verified step by step guidance

Start by factoring the denominators of both rational expressions to find their prime factors. For the first denominator $x^2 - x - 6$, look for 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 $\left(x - 3\right)\left(x - 1\right)$.

The least common denominator (LCD) is the product of all unique factors from both denominators. Combine $\left(x + 2\right)$, $\left(x - 3\right)$, and $\left(x - 1\right)$ to get the LCD: $\left(x + 2\right)\left(x - 3\right)\left(x - 1\right)$.

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Struggling with Beginning 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:
$\frac{3 x^{2} - 12}{x^{2} - x - 6}$ , $\frac{5 x + 10}{x^{2} - 4 x + 3}$