Grade 8 Math Q1 Ep 6 Simplifying Rational Algebraic Expressions

Summary

In this educational video by Deaf Ed TV, Teacher Joshua guides viewers through the process of simplifying rational algebraic expressions, a crucial skill for Grade 8 mathematics. The video explains how to identify and simplify fractions, using factoring to reduce expressions to their simplest form. It covers various examples, including monomials and polynomials, and emphasizes the importance of recognizing when an expression is already in its simplest form. The lesson concludes with a quiz to test understanding, reinforcing the concept that mathematics is about critical thinking and problem-solving.

Takeaways

• 📘 The video is an educational session on simplifying rational algebraic expressions for grade 8 mathematics, led by Teacher Joshua.
• 🍕 An example uses a pizza divided into eight slices to illustrate fractions, where taking two slices leaves six, simplifying to three-fourths.
• 🔢 Simplifying fractions involves writing them in the lowest terms by dividing out common factors from the numerator and denominator.
• 🧩 The video explains that rational algebraic expressions are like fractions, with both the numerator and denominator being polynomials.
• 🔍 To simplify expressions, common factors in the numerator and denominator are identified and divided out.
• 🌟 The video uses the example of simplifying \( \frac{28x^3}{7x^4} \) to \( \frac{4}{x} \) by factoring and dividing common terms.
• ✅ The concept of relatively prime is introduced, where if the numerator and denominator share no common factors, the expression is already in simplest form.
• 👨‍🔬 Albert Einstein's quote about simplicity is mentioned, emphasizing the importance of not oversimplifying to the point of losing meaning.
• 📚 The video provides a step-by-step guide on simplifying rational expressions, including factoring and dividing common terms.
• 📉 The session concludes with a quiz to test the viewer's understanding of simplifying rational algebraic expressions, with five questions and their solutions provided.

Q & A

What are rational algebraic expressions?

Rational algebraic expressions are fractions where both the numerator and the denominator are polynomials, with the denominator not equal to zero.

How can you determine if a rational expression is undefined?

A rational expression is undefined when the denominator is equal to zero. You find the excluded values by setting the denominator to zero and solving for the variable.

What is the fraction of pizza slices left if two out of eight slices are taken?

If two out of eight slices of pizza are taken, six slices are left, which is represented by the fraction 6/8.

How can you simplify the fraction 6/8?

The fraction 6/8 can be simplified by dividing both the numerator and the denominator by their greatest common factor, which is 2, resulting in the simplified fraction 3/4.

What is the simplified form of the expression 28x^3 over 7x^4?

The expression 28x^3 over 7x^4 can be simplified by factoring out common factors. The simplified form is 4/x after factoring and dividing the common factors.

How can you factor the numerator and denominator of the expression 3x - 12 over 5x - 20?

The numerator 3x - 12 has a greatest common factor of 3, and the denominator 5x - 20 has a greatest common factor of 5. Factoring these out and simplifying gives the expression in its simplest form as 3/5.

What does it mean for the numerator and denominator of a rational expression to be relatively prime?

The numerator and denominator of a rational expression are relatively prime if they have no common factors, which means the expression is already in its simplest form and cannot be simplified further.

Who is Albert Einstein and what is his famous equation?

Albert Einstein is a renowned scientist known for his contributions to physics and mathematics, including the theory of relativity. His famous equation is E=mc^2, which relates energy (E) to mass (m) and the speed of light (c).

What is the simplest form of the expression a^3 + b^3 over a^2 - b^2?

The simplest form of the expression a^3 + b^3 over a^2 - b^2 is a^2 - ab + b^2 over a - b, after factoring the numerator as a sum of cubes and the denominator as a difference of squares.

How can you determine if a rational expression is in its simplest form?

A rational expression is in its simplest form if the numerator and denominator are relatively prime, meaning they have no common factors, or if all common factors have been divided out.

What is the process for simplifying a rational algebraic expression?

To simplify a rational algebraic expression, first write the numerator and denominator in factored form, then divide out any common factors. If there are no common factors, the expression is already in its simplest form.

Outlines

00:00

📘 Introduction to Simplifying Rational Algebraic Expressions

Teacher Joshua begins the lesson by welcoming students to Deaf Ed TV, focusing on enhancing mathematical skills for grade 8. The session aims to simplify rational algebraic expressions, which are fractions with polynomial numerators and denominators. The teacher explains how to identify values that make these expressions undefined by setting the denominator to zero. Using a pizza analogy, the concept of fractions is introduced, emphasizing the importance of writing fractions in their simplest form by dividing common factors in the numerator and denominator. The lesson continues with simplifying expressions like 28x^3 over 7x^4 using prime factorization, demonstrating how to identify and divide common factors to achieve the simplest form.

05:00

🔍 Simplifying Rational Expressions Using Factoring

This segment delves deeper into simplifying rational expressions by factoring. The teacher illustrates how to factor out the greatest common monomial factor from both the numerator and the denominator. An example is provided where 3x - 12 over 5x - 20 is simplified to 3/5 by factoring out common terms. The concept of relatively prime terms is introduced, explaining that if the numerator and denominator share no common factors, the expression is already in its simplest form. The segment concludes with a reference to Albert Einstein, emphasizing the importance of simplicity in understanding complex concepts.

10:03

🧩 Factoring Techniques for Rational Expressions

Teacher Joshua continues the lesson by focusing on factoring techniques essential for simplifying rational expressions. The discussion includes the factoring of sums and differences of cubes and squares. The teacher guides students through the process of factoring expressions like a^3 + b^3 over a^2 - b^2, demonstrating how to use specific factoring formulas. The lesson reinforces the importance of recognizing and applying these patterns to simplify expressions effectively.

15:10

📐 Practical Examples and Simplification of Rational Expressions

In this part, practical examples are used to solidify the concept of simplifying rational expressions. The teacher presents an expression, x^2 - 6x + 9 over x^2 - 8x + 15, and guides students through the factoring process. The emphasis is on identifying perfect square trinomials and general trinomials, and factoring them accordingly. The lesson concludes with a summary of the steps to simplify rational algebraic expressions, highlighting the importance of factoring both the numerator and the denominator and dividing out common factors.

20:13

📝 Assessment and Recap of Rational Expression Simplification

The final segment is an assessment and recap of the lesson. Teacher Joshua presents a series of questions to test students' understanding of simplifying rational expressions. The questions cover various scenarios, including identifying simplified fractions and rational expressions, and simplifying complex expressions. The teacher provides detailed explanations for each question, reinforcing the lesson's key points. The segment concludes with a reminder of the importance of practice in mastering mathematical skills and a preview of upcoming lessons on multiplication and division of rational expressions.

Keywords

💡Rational Algebraic Expressions

Rational algebraic expressions are fractions where both the numerator and the denominator are polynomials, and the denominator is not zero. These are a central theme in the video, as they are the primary mathematical objects being simplified. The video discusses how to identify values that make these expressions undefined by setting the denominator to zero. An example from the script is simplifying '28x^3 over 7x^4', where the common factors are identified and simplified.

💡Simplified Form

A simplified form of a fraction or an algebraic expression is when it is written in its lowest terms, meaning there are no common factors between the numerator and the denominator that can be further divided. The video emphasizes the importance of simplifying rational expressions to their simplest form, which makes them easier to understand and work with. For instance, the fraction of pizza slices left after someone takes two slices is simplified from '6/8' to '3/4'.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. It is a key technique used in simplifying rational algebraic expressions, as demonstrated in the video when simplifying '3x - 12 over 5x - 20' by factoring both the numerator and the denominator. Factoring helps in identifying common factors that can be canceled out to simplify the expression.

💡Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. In the context of the video, prime factorization is used to simplify monomials within rational expressions. For example, the number 28 in the numerator is factored into 2 x 2 x 7, which helps in identifying common factors with the variable part of the expression.

💡Greatest Common Monomial Factor (GCMF)

The Greatest Common Monomial Factor is the largest monomial that divides all terms of a polynomial. The video uses GCMF to simplify expressions by factoring out the common parts from both the numerator and the denominator. For example, in simplifying '3x - 12 over 5x - 20', the GCMF of the numerator is 3, and of the denominator is 5.

💡Relatively Prime

Numbers or expressions are relatively prime if they have no common factors other than 1. In the video, the concept is used to determine if a rational expression is already in its simplest form. If the numerator and denominator are relatively prime, no further simplification is possible, as illustrated with the example 'two a plus four over three a minus six'.

💡Sum of Two Cubes

The sum of two cubes is a specific algebraic identity that can be factored into a binomial and a trinomial factor. The video mentions this when simplifying 'a^3 + b^3 over a^2 - b^2', where the numerator is recognized as a sum of two cubes and factored accordingly.

💡Difference of Two Squares

The difference of two squares is another algebraic identity that can be factored into the product of two binomials. The video uses this concept when simplifying the denominator 'a^2 - b^2' by factoring it into '(a + b)(a - b)', which is a standard approach in algebra for such expressions.

💡Perfect Square Trinomial

A perfect square trinomial is a polynomial that can be expressed as the square of a binomial. The video identifies 'x^2 - 6x + 9' as a perfect square trinomial, which factors into '(x - 3)^2'. This recognition allows for further simplification of the rational expression it is part of.

💡General Trinomial

A general trinomial is a three-term polynomial that does not fit the pattern of a perfect square trinomial or a difference of squares. In the video, the denominator 'x^2 - 8x + 15' is identified as a general trinomial, which is factored into '(x - 3)(x - 5)' after finding the appropriate factors of 15 that sum to -8x.

Highlights

Introduction to simplifying rational algebraic expressions, which are fractions with polynomial numerators and denominators.

Explanation of how to identify values that make a rational expression undefined by setting the denominator to zero.

Illustration of simplifying fractions using a pizza example, demonstrating how to find common factors.

Guide on simplifying expressions using prime factorization, with an example of 28x^3 over 7x^4.

Emphasis on the importance of simplifying expressions to their lowest terms for clarity and ease of understanding.

Tutorial on factoring and simplifying the expression 3x - 12 over 5x - 20 to its simplest form.

Discussion on the concept of relatively prime in the context of rational expressions.

Example of simplifying the expression 2a + 4 over 3a - 6, highlighting the absence of common factors.

Albert Einstein's quote on simplicity and its relevance to mathematics and problem-solving.

Challenge problem involving the simplification of a^3 + b^3 over a^2 - b^2, using sum and difference of cubes.

Advice on reviewing factoring lessons to improve skills in simplifying rational algebraic expressions.

Simplification of the expression x^2 - 6x + 9 over x^2 - 8x + 15, using perfect square trinomials.

Recap of the steps to simplify rational algebraic expressions, emphasizing factoring and dividing common factors.

Interactive quiz to test understanding of simplified forms of fractions and rational expressions.

Solution to the quiz question involving the simplification of 7 - x over x - 7, demonstrating factoring out negative 1.

Final challenge problem to simplify 15x^3y^3 over 20x^2y^2, using the greatest common factor method.

Conclusion and encouragement to practice and refine mathematical skills, with a preview of upcoming lessons on multiplication and division of rational expressions.

Transcripts

[Music]

[Music]

good day everyone

and welcome back to deaf ed tv i am

teacher joshua

and i will be your guide in sharpening

your skills and enhancing your minds in

order to face the challenges

in grade 8 mathematics so read your

self-learning mojo

your paper and pens with you let us have

a wonderful day of simplifying rational

algebraic expressions

last episode we talked about rational

algebraic expressions

these are fractions whose numerator and

denominator

are polynomials and the denominator

is not equal to zero

furthermore we can identify values

of the known variable that will make the

rational expression

undefined we find the excluded values by

equating the denominator

to zero and solving for the unknown

values

look at this figure it is a pizza

it is divided into eight slices

what if someone took two slices of pizza

how many are left what

fraction is given by this illustration

there are six slices left out of eight

now what do you remember about writing

fractions

fractions can be written in lowest terms

or simplest form

and it can be expressed in lowest terms

by dividing

common factors in the numerator and

the denominator can you factor

six and eight six is equal to

two times three

well eight is equal to two

times two times two

we can see that there is one common

factor

which is two

we divide the common factors and the

result

of two divided by two is just one

so we are left with three

over two times two

or simplifying

three fourths and since the rational

algebraic expression is a fraction

we can also simplify it with the help of

factoring

simplify the expression 28 x

cubed over 7

x raised to 4 what can you say about

this expression

the numerator and the denominator are

both monomials

or only has one term so how can we

simplify these rational expressions

we can use prime factorization to

simplify

this expression for the numerator the

prime factors of 28

are two times two

times seven and since

x is raised to the power of three it can

be written as

x times x times x

how about the denominator 7

is already a prime number so we write it

down

how about x raised to 4 just like the

numerator

we write x as factors by the number

indicated by the exponent what did you

notice

we can see that there are common factors

in the numerator and the denominator and

when we divide common factors

the result will be one

lastly we multiply what is left two

times two

which is four and write the denominator

x hence the expression can be simplified

as 4 over x

now write the expression 3x minus 12

all over 5x minus 20 in simplest form

remember that we can use factoring in

simplifying a rational expression

how can you factor the numerator and

denominator

if you notice the numerator has a

greatest common monomial factor

the greatest common monomial factor of

3x

minus 12 is 3. so how can we write the

numerator in factored form

divide the numerator by the greatest

common factor 3.

so the numerator is equivalent to three

times the quantity of x minus four

how about the denominator the greatest

common factor of the denominator

is five can you give the complete

factored form of the denominator

it is five times the quantity

x minus four

observe that the numerator and

denominator have

common factors it is x

minus 4 so dividing the common factor

what is the final answer the simplest

form

of 3x minus 12 all over

5x minus 20 is equal

to three-fifths do you remember what

relatively prime means

look at this example simplify

two a plus four all over

three a minus six

the numerator two a plus four has a

greatest common monomial factor

of two and can be written as two

times the quantity k plus two

what is the greatest common monomial

factor in the denominator

it is three so the denominator can be

written

as 3 times the quantity

of a minus 2.

what did you observe do they have common

factors

no they do not it means

that the numerator and denominator are

relatively prime

and if this happens the given rational

expression

is already in its simplest form

do you know who this is this

is albert einstein he is a renowned

scientist

known for his contributions in physics

and mathematics

his famous works include the theory of

relativity

and the equation e is equal to m

c squared which describes the relation

of the kinetic energy

e equal to m

the mass of an object multiplied by the

square of

c the speed of light

one of his quotes also says that

everything must

be made as simple as possible

but not simpler

what do you think does it mean like

fractions

if there are other means to show or

represent these numbers

we can use its simplest form so it will

be

easier to understand in real life

we seek for solutions to our problems

and we tend to go down the easy or the

simple path

but life is not always a simple straight

path

it is full of twists and turns

there are instances that things cannot

be reduced

to something simpler because they may

lose meaning

or importance we need to learn the basic

and the complex so we can create meaning

about the knowledge and skills

that we learn now let us try this

example

write the expression a cubed plus b

cubed

all over a squared minus

b squared in its simplest form

do you remember these polynomials

the numerator a cubed plus b cubed

is a sum of two cubes

while the denominator is a difference of

two squares

to simplify this we need to remember

how to write them in factored form which

we learned

in our previous episodes can you give me

their factored forms

the factors of a sum of two cubes

consist of a binomial factor

and a trinomial factor the binomial

factor of this

special product is the sum of its roots

a plus b what about its trinomial factor

it is the square of the first term a

squared

then what is the next operation

this should be opposite the sign of the

binomial factor

hence it is a minus

then the product of the terms a times b

is a b the last sign is

always positive or a plus sign

and the last term of the trinomial is

b squared how about the denominator

this is a difference of two squares so

what is

its factored form a squared minus b

squared can be expressed as the product

of

a plus b and a

minus b we can divide a plus b

so what is the final answer a cubed

plus b cubed all over a squared minus b

squared

in simplest form is a squared

minus a b plus b squared

all over a minus b

[Music]

if you find the examples difficult i

suggest that you review our lessons in

factoring

mastering that skill will make

simplifying rational

algebraic expressions easy and well

simple let us have x

squared minus 6x plus 9

all over x squared

minus 8x plus 15

to simplify the rational expression we

need to factor the numerator

and denominator what type of polynomial

is in the numerator observe

the first and last terms are perfect

squares

of x and 3 respectively

next 2 times x times 3

is 6x which is equal to the middle term

hence this is a perfect square trinomial

since the middle term of the trinomial

is negative

we write the factor as the square of

x minus 3.

how about the denominator

is the denominator a perfect square

no it is not since the last term 15

is not a perfect square hence

this is a general trinomial we will

multiply the first

and last terms x squared and positive

15.

we will get positive 15 x squared

can you think of factors of 15 x squared

whose sum is negative 8x

the factors must be negative 3 x

and negative 5 x

so the denominator will become the

quantity of x minus 3

times the quantity of x minus 5.

what can you concur since the square

of x minus 3 is just x minus 3

multiplied by itself we can divide

the common factors so what will be the

simplest form of

x squared minus six x plus nine

all over x squared minus eight

x plus fifteen the final answer

will be x minus three

all over x minus five

and that's it for today we have written

rational algebraic expressions in their

simplest form

now let us have a recap rational

algebraic expressions

are fractions its numerator and

denominator

are polynomials and like fractions they

can also be

simplified to simplify a rational

algebraic expression

first we write the numerator and

denominator in factored form

then we divide common factors if there

are remaining factors

we multiply them and if the numerator

and denominator are relatively prime

the rational expression is already

simplified

now that we have finished the lesson it

is important to evaluate

and see what you have learned ready your

paper and pen

and analyze each question carefully

i will give you five seconds to answer

each item

you should also solve the problem with

me as i guide you along the way

number one which of the following

fractions is expressed in simplified

form

is it a two fourths b

four twelfths or c

five sixteenths

the correct answer is c 5

16 because two fourths

can be written as one half

and four-twelfths is equal to one-third

number two which of the following is a

rational expression

in simplest four a

two y over four x

b x squared plus one

all over x minus one

or c x minus one all over

x cubed minus 1.

[Music]

for a rational expression be in its

simplest form

the numerator and denominator should be

relatively prime

in letter a both expressions are

divisible by 2

thus can be further simplified as y

over 2x

for letter c the denominator is a

difference of two cubes

it will be x minus one all over the

quantity

x minus one times the quantity

x squared plus x plus one

which can be further simplified as one

over x squared plus x plus one

so we are left with b x squared plus one

all over x minus one

and we are sure that this is in its

simplest form

since the numerator is a prime

polynomial

number three which of the following is

equivalent to seven

minus x all over x

minus seven is it a

one b 0

or c negative 1.

[Music]

what did you notice with the numerator

and denominator

both show subtraction but the terms

are switched we can rewrite the

expression

as negative x plus seven

all over x minus seven

remember that we include the signs when

we arrange the terms in a polynomial

now what can we do to simplify the

expression

we can factor out negative 1 from the

numerator

and we will have negative 1 times the

quantity

x minus 7 all over

x minus seven then divide common factors

so what do you think will be the

simplified form of this rational

expression

it is c negative one

number four write the rational

expression fifteen

x cubed y cubed over twenty

x squared y squared in simplest four

is it a three x y over four

b three over four x y

or c three fourths

to simplify this expression we can

factor the greatest common factor

what do you think is it the greatest

common factor of the numerator

and denominator is 5 x squared

y squared and the other factors of the

numerator and denominator

are 3xy and 4

respectively so what is our final answer

dividing the common factors the final

answer

is three xy over four

so it is letter a

last item number five which of the

following is the simplest form

of four x plus twelve all over

x squared minus four x

minus twenty one is it a

x minus seven all over four

b four over x minus seven

or c four over x plus seven

we can simplify this rational expression

by getting the greatest common factor

of the numerator and factoring the

general trinomial

in the denominator the numerator can be

expressed

as four times the quantity

of x plus three quick tip

sometimes when we factor polynomials in

rational expressions

we stumble with pieces or parts of the

whole solution

since the denominator is a general

trinomial

a prospect factor in this is the

expression

x plus 3 from the numerator

so what could be the other factor of x

squared minus four x

minus twenty one

what are factors of negative twenty one

such that their sum

is negative four

the factors must be positive three and

negative seven so the denominator is the

quantity

x plus three times the quantity

of x minus seven

which of the choices is the correct

answer

we divide the common factors x plus

three

and we will be left with four over

x -7 so the correct answer

is letter b

did you get the exercise very good

if not i suggest that you review

factoring

and keep on polishing your skills with

the examples and assessment

found on your self-learning module

remember

that in mathematics practice makes you

better

i hope that you have learned a lot in

our episode today

note that rational algebraic expressions

are just fractions

you must focus on patterns you have

observed

in the process in simplifying the

expressions

with hard work and determination i

believe that you can ace

any lesson in mathematics for our next

episode

we will perform multiplication and

division

on rational algebraic expressions

remember math is not only about numbers

and operations

it is an exercise for our minds for us

to be critical

logical and responsible thinkers again

this is teacher joshua reminding you to

keep safe

have a nice day and see you next time

[Music]

do

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you