Multiplying Rational Expressions

Summary

This educational video script offers a comprehensive guide on multiplying and simplifying rational expressions. It begins with a step-by-step demonstration of factoring and canceling terms in a given expression, highlighting the importance of identifying and extracting the greatest common factor (GCF). The script then moves on to factoring trinomials and using the difference of squares technique. It further illustrates how to handle expressions with leading coefficients other than one by multiplying them with the constant term. The video also addresses the identification of excluded values, crucial for avoiding division by zero. Finally, it demonstrates the process of factoring a difference of perfect cubes and concludes with a complete factorization and simplification of a complex rational expression.

Takeaways

• 📘 Multiplying rational expressions involves converting them into fractions and then multiplying the numerators and denominators accordingly.
• 🔍 Factoring expressions completely is crucial before multiplying to simplify the process and identify common factors for cancellation.
• 🧩 The greatest common factor (GCF) is used to simplify expressions by dividing out common factors from the numerator and denominator.
• ✂️ Cancelling common factors between the numerator and denominator is a key step in simplifying rational expressions.
• 🔢 Factoring trinomials involves finding two numbers that multiply to the constant term and add up to the middle coefficient.
• 🔄 The difference of squares technique is used to factor expressions of the form a^2 - b^2 into (a + b)(a - b).
• 📏 Factoring by grouping is a method used for expressions where the coefficients follow a pattern that allows for common factors to be extracted.
• 📉 Identifying excluded values, or points of discontinuity, is important as they represent values that make the denominator zero and are not valid for the expression.
• 🔑 The difference of cubes formula a^3 - b^3 = (a - b)(a^2 + ab + b^2) is used to factor expressions involving the difference of two cubes.
• ✅ Simplifying rational expressions results in a final expression that is easier to work with and understand.

Q & A

What is the first step when multiplying rational expressions?

The first step is to ensure that the expressions are in fractional form and then to factor them completely.

How do you simplify the expression 7x + 14 divided by x^2 - 8?

First, factor out the greatest common factor (GCF) of 7 from the numerator to get 7(x + 2). Then factor the denominator using the difference of squares to get (x + 2)(x - 2). After canceling out common factors, the simplified form is x + 2.

What technique is used to factor x^2 - 4?

The difference of perfect squares technique is used, which results in (x + 2)(x - 2).

What are the excluded values when simplifying rational expressions?

Excluded values are the values of the variable that make any denominator zero. These values are identified by setting each factor in the denominator equal to zero.

How do you factor the trinomial 10x^3 - 70x^2 + 120x?

First, take out the GCF of 10x, resulting in x^2 - 7x + 12. Then, factor by grouping or by finding two numbers that multiply to 24 (2*12) and add to -7 (-8 + 3), which are -8 and 3. The factored form is (x - 3)(x + 4).

What is the process for factoring a difference of perfect cubes?

The formula for factoring a difference of perfect cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2). Identify 'a' and 'b' such that a^3 is the first term and b^3 is the last term of the cube difference.

How do you simplify the expression 5x^2 - 15x divided by 2x^2 + 11x + 12?

First, take out the GCF of 5x from the numerator. Then, factor the quadratic in the denominator by finding two numbers that multiply to 24 and add to 11, which are 8 and 3. The simplified form is x - 3 over 2(x + 4).

What is the final simplified form of the expression 7x + 14 divided by x^2 - 8 multiplied by x^2 + 3x - 10?

After factoring and canceling, the final simplified form is 7(x + 5)/2.

How do you identify the points of discontinuity in a rational expression?

Points of discontinuity are identified by finding the values of the variable that make any factor in the denominator equal to zero, as these values would make the expression undefined.

What is the final simplified form of the expression 3x^3 - 24 divided by 2x^2 - 14x + 20 times 4x^3 - 20x^2 + 3x - 15 divided by x^2 + 6x + 12?

After factoring and canceling, the final simplified form is 4x^2 + 3/2.

Outlines

00:00

📘 Multiplying and Simplifying Rational Expressions

This paragraph introduces the process of multiplying and simplifying rational expressions. The first example involves multiplying \( \frac{7x+14}{x^2-8} \) by \( \frac{x^2+3x-10}{1} \). The process starts by factoring the expressions and identifying the greatest common factor (GCF). The GCF of 7x+14 is 7, which simplifies to \( \frac{x+2}{x^2-4} \). The denominator \( x^2-4 \) is factored using the difference of squares technique into \( (x+2)(x-2) \). The numerator \( x^2+3x-10 \) is factored into \( (x+5)(x-2) \). After canceling common factors, the final simplified form is \( \frac{7(x+5)}{2} \). The paragraph also explains how to identify excluded values, which are the values that make the denominator zero and thus are not in the domain of the function.

05:00

📗 Factoring Techniques and Simplifying Expressions

The second paragraph delves into more examples of multiplying and simplifying rational expressions. It begins with the expression \( \frac{5x^2-15x}{2x^2+11x+12} \times \frac{3x^2-48}{10x^3-70x^2+120x} \). The process involves factoring out the GCFs and then simplifying the expressions. The numerator \( 5x^2-15x \) simplifies to \( x(3x-3) \) after factoring out the GCF of 5x. The denominator \( 2x^2+11x+12 \) is factored by grouping, resulting in \( 2(x+4)(x+3) \). The second part of the expression is simplified by factoring out the GCF of 10x and then factoring the resulting trinomial. The final simplified form after canceling common factors is \( \frac{3}{2(2x+3)} \). The paragraph also discusses how to identify excluded values by setting factors equal to zero and solving for x.

10:02

📙 Advanced Factoring and Canceling in Rational Expressions

The third paragraph presents a more complex example of multiplying and simplifying rational expressions, involving the expression \( \frac{3x^3-24}{2x^2-14x+20} \times \frac{4x^3-20x^2+3x-15}{x^2+6x+12} \). The process starts by factoring out the GCFs and then using advanced factoring techniques such as the difference of cubes and factoring by grouping. The numerator \( 3x^3-24 \) simplifies to \( x(x^2-8) \) after factoring out the GCF of 3. The denominator \( 2x^2-14x+20 \) simplifies to \( 2(x^2-7x+10) \) by dividing each term by 2. The expression \( x^2-7x+10 \) is then factored by grouping into \( (x-5)(x-2) \). After canceling common factors, the final simplified form is \( \frac{4x^2+3}{2} \). The paragraph also emphasizes the importance of canceling all possible factors and identifying excluded values to ensure the solution is fully simplified and valid.

Keywords

💡Rational Expressions

Rational expressions are mathematical expressions that involve ratios of two polynomials. In the video, the theme revolves around multiplying and simplifying rational expressions, which is a critical skill in algebra. The script mentions multiplying rational expressions like '7x plus 14, divided by, x squared, minus eight' by another rational expression, demonstrating the process of simplifying these expressions by factoring and canceling common factors.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. It plays a central role in the video as it is used to simplify rational expressions. For instance, the script describes factoring '7x plus 14' by taking out the greatest common factor (GCF) of 7, which simplifies the expression to 'x plus 2'.

💡Greatest Common Factor (GCF)

The GCF is the largest number or polynomial that divides two or more numbers or expressions without leaving a remainder. In the video, finding the GCF is a recurring step in simplifying rational expressions. The script illustrates this by taking the GCF out of '7x plus 14' and '2x squared minus 8' to simplify the expressions before multiplying them.

💡Simplifying

Simplifying in the context of the video refers to the process of making rational expressions easier to understand and work with by reducing them to their most basic form. This is done through factoring and canceling out common factors. The script provides examples of simplifying expressions like '5x squared minus 15x' by factoring out the GCF and then canceling common factors.

💡Difference of Perfect Squares

This is a specific algebraic technique used to factor expressions of the form 'a squared minus b squared'. In the video, the script uses this technique to factor 'x squared minus 4' into '(x plus 2)(x minus 2)', which is a key step in simplifying the given rational expressions.

💡Trinomial

A trinomial is a polynomial with three terms. The video discusses factoring trinomials, which is essential for simplifying rational expressions. The script provides an example of factoring a trinomial 'x squared plus 3x minus 10' by finding two numbers that multiply to -10 and add to 3, resulting in '(x plus 5)(x minus 2)'.

💡Canceling

Canceling in algebra refers to the process of removing common factors from the numerator and denominator of a fraction. This is a key step in simplifying rational expressions, as demonstrated in the video. The script shows canceling out 'x plus 2' and 'x minus 2' from the numerator and denominator to simplify the expression.

💡Factor by Grouping

Factor by grouping is a method used to factor expressions, particularly when the expression is not easily factorable by other means. The video script describes using this method to factor '2x squared plus 11x plus 12' by grouping terms and finding common factors, which simplifies the expression to '(x plus 4)(2x plus 3)'.

💡Excluded Values

Excluded values are the values of the variable that make the denominator of a rational expression equal to zero and thus are not allowed in the domain of the expression. The video emphasizes identifying these values to ensure the validity of the simplification process. The script mentions that 'x cannot equal negative 4, 4, 0, and 3 and negative 3 over 2' as these would make the denominator zero.

💡Difference of Perfect Cubes

This is an algebraic identity that allows for the factoring of expressions of the form 'a to the third minus b to the third'. The video script uses this identity to factor 'x cubed minus 8' into '(x minus 2)(x squared plus 2x plus 4)', illustrating a more advanced technique for simplifying rational expressions.

Highlights

Introduction to multiplying and simplifying rational expressions

Example of multiplying 7x + 14 by x^2 + 3x - 10

Factoring out the greatest common factor (GCF) of 7 from 7x + 14

Simplifying 2x^2 - 8 by factoring out 2

Factoring the trinomial x^2 - 4 into (x + 2)(x - 2)

Cancellation of common factors (x + 2) and (x - 2) in the expression

Final simplified form of the first example as 7(x + 5) / 2

Second example involving 5x^2 - 15x and 3x^2 - 48

Factoring out the GCF of 5x from 5x^2 - 15x

Factoring 3x^2 - 48 using the difference of squares technique

Factoring the cubic polynomial 10x^3 - 70x^2 + 120x

Identifying the excluded values for the rational expression

Third example with 3x^3 - 24 and 2x^2 - 14x + 20

Factoring by grouping in the expression 4x^3 - 20x^2 + 3x - 15

Using the formula for the difference of perfect cubes to factor x^3 - 8

Final simplified form of the third example as 4x^2 + 3 / 2

Transcripts

in this video we're going to focus on

multiplying rational expressions

and simplifying them as well

so here's the first example

7x plus 14

divided by

x squared

minus eight

and let's multiply it by

x squared

plus three x

minus ten

feel free to try this example

now the first thing i would do is

put the second part into a fraction

write it over one

now what you want to do is you want to

factor completely and cancel

7x plus 14 has a gcf of 7 so we want to

take that out

7x divided by 7 is x

14 divided by 7 is 2.

now 2x squared minus 8 we can also take

out the greatest common factor which is

2

and that's going to leave behind x

squared and negative eight divided by

two is negative four

on the right we have a trinomial with a

leading coefficient of one

so we need to find two numbers that

multiply to negative ten but add to

three

so that's positive five and negative two

so this is going to be x plus five

and x minus two

now let's factor x squared minus four

using the difference of perfect squares

technique we know it's going to be

the square root of x squared is x the

square root of 4 is 2

and so it's going to be x plus 2 and x

minus 2.

now let's see what we can cancel

so we can cancel an x plus two

and we can cancel an x minus two

so we have a seven left over

and x plus five

and a two

so the final answer is let's see if i

can fit it in here

seven

times x plus five

divided by two

so that's the solution to this problem

fully simplified

now let's try another example

5x squared

minus 15x

divided by

2x squared

plus 11x plus 12

multiplied by

3x squared minus 48

divided by

10 x cubed

minus 70

x squared

plus

120x

so go ahead and simplify factor

everything

and cancel

so 5x squared minus 15x we can take out

the gcf

which is 5x

5x squared divided by 5x is x

negative 15x divided by 5x

is negative 3.

now 3x squared minus 48

we can take out a 3

which will give us x squared minus 16.

and

we can factor that using the difference

of squares technique

the square root of 16 is 4.

so we're going to have x plus 4 and x

minus four

now what about ten x cubed

minus seventy x squared plus one twenty

x

well we can begin by taking out the gcf

which is 10x

so we're going to have x squared

negative 70x squared divided by 10x

is negative 7x

120x divided by 10x is 12.

two numbers that multiply to 12 but add

to negative seven

are negative three and negative four

so this is going to be x minus three

and x minus four so let's go ahead and

put that here

and so this is what we have

let's just get rid of this

now let's focus on the last part

2x squared plus 11x

plus 12.

we have a trinomial with the leading

coefficient being something other than

one it's two

so when it's not one

multiply the leading coefficient by the

constant term

so two times 12 which is 24.

now what two numbers multiply to 24

but add to the middle coefficient 11.

this is going to be 8 and 3. 8 plus 3 is

11. 8 times 3 is 24. so we're going to

do is

we're going to replace 11x

with 8x plus 3x

and then factor by grouping

so in the first two terms take out the

gcf

the gcf is 2x

and so we'll be left with x plus four

and in the last two terms take out the

greatest common factor

which is three

and we're going to get x plus four

so we're gonna have two parentheses one

of which will contain x plus four

and the other one it's going to have the

stuff on the outside the two x plus

three

and so that's how we can factor it

so it's x plus four times two x plus

three

so now let's simplify

what can we cancel

we can cancel an x minus three

and we can cancel an x minus four

in addition to that we can get rid of x

plus four

and also we can take out an x

and also we can reduce five over ten

five over ten reduces to one over two

ten divided by five is two

so what we have left over

is a three

a two and a two x plus three

so the final answer is three divided by

two times

two x plus three

and so that is the solution

sometimes you need to identify

the excluded values

and here's the basic idea

you cannot have a zero in the

denominator

so four

will produce a zero in the denominator

even though it cancels and so will three

zero if you just see x

negative four

and also if you set two x plus three

equal to zero

you should get negative three over two

so that's how you can identify the

excluded values so we have negative 4

and 4

0 and 3 and negative 3 over 2.

those are the points of discontinuity

x cannot equal any of those numbers

now let's work on another problem

3x q minus 24

divided by

2x squared

minus 14x

plus

20. times 4x cubed

minus 20x squared

plus

3x

minus 15

divided by

x squared

plus six x plus twelve

so pause the video try this problem

now the first thing we need to do is

look for any gcfs

and notice that both numbers on top are

divisible by three so we can take out a

three

three x cubed divided by three is x

cubed

negative 24 divided by three is negative

eight

now in the denominator we all have even

numbers which means that we can divide

each of those by two

and so we'll be left with x squared

minus seven x plus ten

here all the coefficients are divisible

by three

so if we take out a three it's going to

be x squared plus

two x

plus four

now for this expression

notice that we can factor it by grouping

negative 20 divided by 4 is negative 5.

negative 15 divided by 3 is also

negative 5.

so if the first two coefficients have

the same ratio as the last two

coefficients

you can factor by grouping

so let's take out the gcf in the first

two terms

which is going to be 4x squared

4x cubed divided by 4x squared is x

negative 20x divided by 4x squared

is negative 5.

now in the last two terms

we can take out a 3.

3x divided by 3 is x

negative 15 divided by 3 is

negative 5.

now let's start with this

how can we factor x cubed minus eight

here's the formula that we need to use

a to the third minus b to the third

is going to be a minus b

times a squared

plus a b

plus a b squared

so if a to the third

is x to the third

a has to be x

and if b to the third is eight b has to

be the cube root of eight which is two

so a is x b is two so a squared is going

to be x squared

and then a b that's

x times negative two

well actually b is positive 2.

this negative sign is already there so

a times b is going to be x times 2 which

is positive 2x

there's a plus sign

and then b squared that's 2 times 2

which is 4.

so that's how you can factor

a difference of

perfect cubes

now let's factor the trinomial two

numbers that multiply to 10 but add to

negative seven

that's going to be negative two and

negative five

so this becomes x minus five

and x minus two

now we can't really factor

x squared plus two x plus four but we

don't need to because it will be

canceled soon

so i'm just gonna rewrite it for now

now going back to this expression

we need to take out the x minus five

since it's common to both terms

and

we're gonna have four x squared plus 3

left over

so now the rational expression is

completely factored

now let's cancel

so we can cancel x minus 2.

we can cancel x squared plus two x plus

four

we can cancel x minus five

and

we can cancel a three

so the only things that we have left

over

are those two so the final answer is 4x

squared plus 3

divided by 2

and that's it

you