Find the least common denominators of the rational expressions: 2 x 2 + 5 x − 3 x 2 − 9 , 4 x − 12 x 2 − x − 6 , x 2 + x − 2 x 2 − 4 x + 3 \frac{2x^2+5x-3}{x^2-9},\frac{4x-12}{x^2-x-6},\frac{x^2+x-2}{x^2-4x+3}

This problem asked us to find the least common denominator of the rational expressions, and we're now moving on to part c. Now this problem looks a little bit complicated because I can see that we are now dealing with three rational expressions. But don't sweat it because we can solve this exactly like how we've solved the other problems when we're trying to find the least common denominator. So the same way we find the LCD for two rational expressions, we can use the same strategy for three, where I really just need to factor all the denominators that I see. I'm going to start with this first rational expression, two x squared plus five x minus three. In the denominator, notice we have x squared minus nine, which is the same thing as x squared minus three squared. Nine can be written as three squared. Now, of course, this will break down into a difference of squares. So this is the same thing as x plus three times x minus three. And next, I'll move on to the second rational expression where we have four x minus 12 in the numerator and x squared minus x minus six in the denominator. Well, I can see here that we have a quadratic polynomial with one as the leading coefficient. So I'm looking for a pair of numbers that multiplies to give me this negative six and adds to give me this negative one. Well, thinking about that, that's going to be two and negative three. Two times negative three is negative six. Two plus negative three is negative one. So we're going to get x plus two times x minus three. Now lastly, I'll move to this last rational expression where we have x squared plus x minus two in the numerator. And then in the denominator, we wanna factor this rational or we wanna factor this denominator. But I can see that we, once again, have a quadratic polynomial with one as the leading coefficient in front of x squared, meaning I can just think of a pair of numbers that multiplies to give me positive three and adds to give me negative four. Well, if I think about it, that's going to be negative one and negative three. Those multiply to give me positive three and add to give me negative four, giving me x minus one times x minus three. Now let's find the least common denominator. Well, I can see that we have an x plus three. I don't see an x plus three anywhere else in any of these denominators, so that x plus three is a unique prime factor. Now let's move on to this x minus three. I can see that x minus three is a common term that we have in all three of these denominators. So x minus three by itself will be a unique prime factor. There's no other x minus three's we need to write, we only need to write it once. And I can also see we have an x plus two. I don't see an x plus two anywhere else, so x plus two is a unique prime factor. Now, I also see that we have an x minus one. I don't see an x minus one anywhere else, so x minus one is a unique prime factor, and this would be the LCD and the solution to part c of this problem. Hope you found this video helpful.

Find the least common denominators of the rational expressions:2x2+5x−3x2−9,4x−12x2−x−6,x2+x−2x2−4x+3\frac{$$2x^2+5x-3$$}{$$x^2-9$$},\frac{4x-12}{$$x^2-x-6$$},\frac{$$x^2+x-2$$}{$$x^2-4x+3$$}

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

(x−3)(x−3)2(x−2)\left(x-3\right)\left(x-3\$$right)^2$$\left(x-2\right)(x−3)(x−3)2(x−2)

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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

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

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Multiple Choice

Find the least common denominators of the rational expressions:
$\frac{2 x^{2} + 5 x - 3}{x^{2} - 9}, \frac{4 x - 12}{x^{2} - x - 6}, \frac{x^{2} + x - 2}{x^{2} - 4 x + 3}$

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

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

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

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

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Verified step by step guidance

Start by factoring each denominator completely to identify their prime factors. For the first denominator $x^2 - 9$, recognize it as a difference of squares and factor it as $\left(x+3\right)\left(x-3\right)$.

Next, factor the second denominator $x^2 - x - 6$. Find two numbers that multiply to $-6$ and add to $-1$. This factors as $\left(x-3\right)\left(x+2\right)$.

Then, factor the third denominator $x^2 - 4x + 3$. Find two numbers that multiply to \$3$ and add to $-4$. This factors as $\left(x-3\right)\left(x-1\right)$.

To find the least common denominator (LCD), take each distinct factor from all denominators, using the highest power of each factor that appears. Here, the factors are $\left(x+3\right)$, $\left(x-3\right)$, $\left(x+2\right)$, and $\left(x-1\right)$. Note that $\left(x-3\right)$ appears in all denominators but only to the first power, so include it once.

Multiply these factors together to write the LCD as $\left(x+3\right)\left(x-3\right)\left(x+2\right)\left(x-1\right)$. This expression represents the least common denominator for the given rational expressions.

6:33m

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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:2x2+5x−3x2−9,4x−12x2−x−6,x2+x−2x2−4x+3\(\frac{2x^2+5x-3}{x^2-9}\),\(\frac{4x-12}{x^2-x-6}\),\(\frac{x^2+x-2}{x^2-4x+3}\)

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Find the least common denominators of the rational expressions:
$\frac{2 x^{2} + 5 x - 3}{x^{2} - 9}, \frac{4 x - 12}{x^{2} - x - 6}, \frac{x^{2} + x - 2}{x^{2} - 4 x + 3}$