Grade 8 Math Q1 Ep 6 Simplifying Rational Algebraic Expressions
Summary
TLDRIn 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
📘 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.
🔍 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.
🧩 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.
📐 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.
📝 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.
Mindmap
Keywords
💡Rational Algebraic Expressions
💡Simplified Form
💡Factoring
💡Prime Factorization
💡Greatest Common Monomial Factor (GCMF)
💡Relatively Prime
💡Sum of Two Cubes
💡Difference of Two Squares
💡Perfect Square Trinomial
💡General Trinomial
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
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