FACTORING PERFECT SQUARE TRINOMIALS || GRADE 8 MATHEMATICS Q1
Summary
TLDRThis educational video script explains how to identify and factor perfect square trinomials. It clarifies that a perfect square trinomial must have a positive leading term that is a perfect square, a middle term twice the product of the square roots of the first and last terms, and a last term that is also a perfect square and positive. The script provides examples to illustrate the process, including incorrect cases, and demonstrates the factoring technique for perfect square trinomials, emphasizing the importance of checking each term's square status and the middle term's relationship to the others.
Takeaways
- 📚 The video explains how to identify and factor perfect square trinomials, which are algebraic expressions that can be written as the square of a binomial.
- 🔍 A perfect square trinomial must have a first term that is a perfect square and always positive, a middle term that is twice the product of the square roots of the first and last terms, and a last term that is also a perfect square and positive.
- 📉 The script provides examples of expressions that are and are not perfect square trinomials, helping to clarify the concept.
- 📝 To factor a perfect square trinomial, one should express it as the square of the sum or difference of two terms, depending on the sign of the middle term.
- 🤔 The video emphasizes the importance of checking if both the first and last terms are perfect squares and if the middle term is twice their product.
- 📐 It demonstrates the process of factoring perfect square trinomials by providing step-by-step examples, such as \(x^2 + 2xy + y^2\) which factors to \((x + y)^2\).
- 🚫 The script clarifies that not all trinomials are perfect squares, and it's crucial to verify the conditions before attempting to factor.
- ✅ The video includes a method to factor expressions that are not perfect square trinomials by first factoring out the greatest common factor (GCF), if applicable.
- 📝 The script gives a clear example of how to handle negative middle terms in perfect square trinomials, such as \(x^2 - 22x + 121\) which factors to \((x - 11)^2\).
- 🔢 The importance of recognizing perfect squares, such as \(4x^2\) being \((2x)^2\) and \(25\) being \(5^2\), is highlighted for successful factoring.
- 👍 The video concludes with an encouragement to like, subscribe, and follow the channel for more educational content.
Q & A
What is the main topic of the video?
-The main topic of the video is identifying and factoring perfect square trinomials.
What is a perfect square trinomial?
-A perfect square trinomial is an algebraic expression that can be written as the square of a binomial, meaning it has the form (ax + by)^2.
What are the characteristics of the first term in a perfect square trinomial?
-The first term in a perfect square trinomial must be a perfect square and it is always positive.
What is the condition for the middle term of a perfect square trinomial?
-The middle term must be twice the product of the square roots of the first and last terms.
What should the last term of a perfect square trinomial be?
-The last term must also be a perfect square and it is always positive.
How can you determine if the expression 4x^2 + 20x + 25 is a perfect square trinomial?
-You can determine it's a perfect square trinomial because the first term (4x^2) and the last term (25) are perfect squares, and the middle term (20x) is twice the product of the square roots of the first and last terms (2x * 5).
Why is the expression x^2 + 5x + 6 not a perfect square trinomial?
-The expression x^2 + 5x + 6 is not a perfect square trinomial because the last term (6) is not a perfect square.
What is the factored form of a perfect square trinomial x^2 + 2xy + y^2?
-The factored form of the perfect square trinomial x^2 + 2xy + y^2 is (x + y)^2.
How do you factor a perfect square trinomial with a negative middle term?
-If the middle term is negative, the factored form is the square of the binomial with a negative sign, such as (x - y)^2.
What is the process for factoring the expression 16x^2 + 72x + 81?
-First, confirm that the first and last terms are perfect squares (16x^2 and 81) and that the middle term (72x) is twice the product of their square roots (4x * 9). Then, factor it as (4x + 9)^2.
Can the expression 27a^2 + 72ab + 48b^2 be a perfect square trinomial?
-No, the expression 27a^2 + 72ab + 48b^2 cannot be a perfect square trinomial because 27 and 48 are not perfect squares.
What is the factored form of the expression 4x^3 - 24x^2 + x, given it is a perfect square trinomial?
-The factored form of the expression 4x^3 - 24x^2 + x is x(x - 6)^2, after factoring out the greatest common factor x.
Outlines
📚 Introduction to Perfect Square Trinomials
This paragraph introduces the concept of perfect square trinomials, explaining that they are expressions that can be factored into the square of a binomial. It provides examples of expressions that are and are not perfect square trinomials, such as 'x squared plus two x squared plus y squared' and 'x squared plus 5x plus 6', respectively. The criteria for identifying a perfect square trinomial are outlined: the first and last terms must be perfect squares and positive, and the middle term must be twice the product of the square roots of the first and last terms.
🔍 Identifying Perfect Square Trinomials
The second paragraph delves deeper into the identification process of perfect square trinomials. It explains the necessity for the first and last terms to be perfect squares and the middle term to be twice the product of the square roots of the first and last terms. Examples given include '4x squared plus 20x plus 25' and '9x squared plus 30xy plus 25y squared', which are confirmed as perfect square trinomials, and '4x squared plus 2xy plus y squared', which is not. The paragraph emphasizes the importance of the middle term matching the required criteria for a trinomial to be considered perfect square.
📐 Factoring Perfect Square Trinomials
This paragraph focuses on the process of factoring perfect square trinomials. It presents the formula for factoring such expressions, which involves taking the square root of the first and last terms and squaring them in the factored form. Examples are provided to illustrate the process, including 'x squared plus 10x plus 25' and '16x squared plus 72x plus 81', which are factored into '(x + 5) squared' and '(4x + 9) squared', respectively. The paragraph also discusses the implications of a negative middle term and how it affects the sign in the factored form.
📘 Advanced Factoring of Perfect Square Trinomials
The final paragraph presents more complex examples of perfect square trinomials, including those with negative middle terms and those that require factoring out a greatest common factor (GCF) before applying the perfect square trinomial formula. Examples like 'x squared minus 22x plus 121' and '25m squared minus 20mn plus 4n squared' are used to demonstrate the factoring process. The paragraph concludes with a reminder to check if an expression is a perfect square trinomial before attempting to factor it and to look for the GCF if necessary.
Mindmap
Keywords
💡Perfect Square Trinomial
💡Factor
💡Middle Term
💡First Term
💡Last Term
💡Positive
💡Binomial
💡Square Root
💡Product
💡Greatest Common Factor (GCF)
Highlights
A perfect square trinomial is a special type of quadratic expression that can be factored into the square of a binomial.
The first term of a perfect square trinomial must be a perfect square and always positive.
The middle term is twice the product of the square roots of the first and last terms.
The last term of a perfect square trinomial must also be a perfect square and positive.
Expressions like x^2 + 2x + y^2 and 4x^2 + 20x + 25 are examples of perfect square trinomials.
x^2 + 5x + 6 is not a perfect square trinomial because 6 is not a perfect square.
The expression 9x^2 + 30xy + 25y^2 is a perfect square trinomial, factoring to (3x + 5y)^2.
4x^2 + 2xy + y^2 is not a perfect square trinomial because the middle term does not match the required form.
To factor a perfect square trinomial, take the square root of the first and last terms and multiply them.
If the middle term is negative, the factored form will be the square of (x - y).
For x^2 + 10x + 25, the factored form is (x + 5)^2, demonstrating the perfect square trinomial property.
16x^2 + 72x + 81 factors to (4x + 9)^2, showing the process of identifying and using perfect square trinomials.
x^2 - 22x + 121 is a perfect square trinomial that factors to (x - 11)^2, with a negative middle term.
25m^2 - 20mn + 4n^2 factors to (5m - 2n)^2, illustrating the application of the perfect square trinomial formula.
27a^2 + 72ab + 48b^2 is not a perfect square trinomial due to the non-perfect square terms.
For expressions that are not perfect square trinomials, factoring involves finding the greatest common factor.
4x^3 - 24x^2 + x factors to x(x - 6)^2, showing the process of factoring by finding a common factor.
The video concludes with a reminder to like, subscribe, and hit the bell for more educational content.
Transcripts
[Music]
in this video
we will identify whether or not an
expression
is a perfect square trinomial and we
will factor
perfect square trinomials completely
so first identify
we have x squared plus two x squared
plus y squared so this is a perfect
square trinomial
later i will explain why and how
next 4x squared plus 20x plus 25 this is
also a perfect squared trinomial
x squared plus 5x plus 6
this is not a perfect square trinomial
9x squared plus 30xy plus 25y
squared this is a perfect square
trinomial
and 4x squared plus 2xy plus y
squared this is not a perfect square
trinomial okay
so how will we know
if the given expression is a perfect
square trinomial
a perfect square trinomial
your first term must be a perfect square
and it must be positive it's always
positive
so tatanda and young first term nothing
but perfect square shape it's not be
nothing perfect square
of course we can get its square root
okay
so young first term nothing
positive next the middle term must be
twice the product of first and the last
term so your middle term nothing
multiplying
negative dependence are given okay
and then next our last term must be a
perfect square
and it must be always positive so unless
terminating just like your first term
so to sum it up the first term and the
last term must be a perfect square and
the positive
uh term and then the middle term not
then it must be twice the product of
first and last term
okay let's have an example so eto london
buckets a perfect square trinomial so
x squared plus 2 x y plus y squared
so this is again a perfect square
trinomial y
so sabi first and last term not in a
perfect square and it must be
positive so as you can see in the given
our first and last term
are both positive and then so it's a
check nothing
perfect squares so first term not in a
perfect square
also our last term or young story term
not m
now that path tag me multiply nothing on
first
and last term at the names not in chasse
2 but the resulting product must be the
middle term so 2
times x times y that is 2xy and that is
our middle term so therefore
this is a perfect square trinomial
next 4x squared plus 20x
plus 25 so let's see
let's see if the our first term and last
term
are both perfect squares so
in 4x squared that is 2x
cassette 2 raised to 2 that is 4 and
then x raised to 2
that is x squared so 4x squared and then
25
we all know that 25 is a perfect square
and that is 5.
so 5 squared is 25. now we will multiply
the first and the last term so that is
2x and 5
so 2 times 2x times 5
so that is 4x times 5 that is 20x
and that makes it a perfect square
trinomial next we have
x squared plus five x plus six so let's
check
so our first term is a perfect square
our last term or our third term
is not a perfect square so six is not a
perfect square so dun palang
masa sabina hindi is a perfect square
trinomial so this is not a perfect
square
trinomial next number four nine x
squared plus 30 x y plus 25y squared
so our first term is a perfect square
that is 3x
and then our third term is a perfect
square that is 5y
and then we will multiply 3x and 5y
so twice so we will have 2 times 3x that
is 6x plus 5y that is 30xy
and then 4x squared plus 2xy plus y
squared so our first term is a perfect
square that is 2x
our last term is a perfect square
also and then we will uh multiply
2x and y and then times 2pa
so we will now have 2 times 2 x
is equal to 4x times y that is 4xy
but as you can see a middle term not
indito i
2x y long so this is not a perfect
square trinomial
so mean
middle term that it must be twice the
product of your first and last term
so that makes it not a
perfect square trinomial
okay now how to factor
perfect square trinomials
x squared plus 2xy plus y squared
that will become x plus y
raised to 2 or simply the quantity of x
plus y
squared or the square of x plus y
all right where x is your first term and
then your y
here is your last term so dul lang tayo
titting in
now take note kapagang trinomial nothing
or a perfect square trinomial nathan i
x squared plus 2 x y kappa middle term
net and a positive
then young resulting factor
is positive then okay
negative your middle term the resulting
factor is
negative then or minus okay so this
is now our factored form so again
x squared plus 2xy plus y squared but
again i'm given a 10 a perfect square
trinomial
the factoid form is the square of x plus
y
and then if it's x squared minus 2xy
plus y
squared the factored form is x
minus y squared or the square of x minus
y
let's have an example so i have here x
squared plus 10 x plus 25
so that pattern factored formula
or the square
perfect square trinomial a perfect
square and first and last term net n so
x and then five so they are both perfect
squares
and then we will check if the middle
term is twice the product of your first
and last term so
multiply nothing on x and five
sa two so we will have two times
x that is two x times five that is ten x
and that satisfies our our middle term
so e
using the square of x plus y because the
given expression is a perfect square
trinomial
so we can now have the factored form
just copy the first term which is x
and copy that last term which is five
that's it as simple as that
next i have here 16x squared plus 72x
plus 81.
so since a middle terminal and i
positive or plus
therefore the resulting factor or young
factory formulating will be
the square of x plus y so plus i said
the middle term is plus
so we will check first if the given
expression
is a perfect square trinomial case in
this a perfect square trinomial we
cannot use this form the square of x
plus y
okay so check nothing on first and last
term nothing come perfect square
ah perfect square sila so 16x squared is
4x
that is a perfect square and 81 is also
a perfect square and that
is 9 squared so now let's check the
middle term
if it is twice the product of 4x
and 9 okay so 2 times 4x that is 8x
times
9 that is 72x and that satisfies our
middle term
okay so therefore the factored form just
copied the first term which is 4x
and then our last term which is nine so
the factored form is
the square of four x plus nine or
uh dalawang binomial four x plus nine
times four x plus nine
okay next
i have here x squared minus 22x plus
121. so as you can see
the factored form must be x minus
y but because our middle term
here is minus so that pattern factored
formula is square
of x minus y
okay so before anything else before you
proceed
let's check first if it's
a perfect square trinomial so
check nothing first and last term come
perfect square so
for our first term is x squared so
obviously it's a perfect square
and our last term is 121 which is 11
squared
okay so therefore
let us now check the middle term if it's
twice
the product of your first and last term
so we will multiply x and 11
times two so 2x times 11 that is 22
x and that satisfies our middle term
so therefore we can now proceed to the
factory form
just follow the rule okay so we will
just copy the first term which is
x and then the second term are the
last term which is 11. so our factored
form is
x minus 11 squared next
25 m squared minus 20 m n
plus 4 n squared is equal to the square
of
x minus y so again this is x minus
minus because our middle term is minus
okay so check first if it's a perfect
square trinomial so again
we need to check kung perfect square
trinomial
term 25 m squared so that is 5m
that is a perfect square and then 4n
squared that is 2n
and 2n is a perfect square and no 4n
squared is a perfect square and that is
2
at the quantity of 2n raised to 2.
and then we will get twice the product
of your first
and last term which is 5 m into n so 2
times 5 that is 10
m times 2 n that is 20 m
n so our factored form will be
we will just copy the first and last
so that is 5m and 2n so our factored
form
is the square of 5m minus
2n or the square of the difference of 5m
and 2n
okay next
i have here 27 a squared plus 72 a b
plus 48 b squared so
now this is an example of an expression
which
is not a perfect square trinomial
because as you can see
27 is not a perfect square also 48 so it
is the square of x plus y
and the square of x minus y if the given
polynomial is a perfect square trinomial
in this a perfect square trinomial all
you have to do is to
factor out okay so pakistan factor out
you need to look for the gcf so since
and gcf need to a3
so if i factor out nothing sha i know
you multiply most of three to get 27a
squared
that is 9a squared okay
and then an imma multiply muscle 3 to
get 72 a b
so that is 24 a b
what will you multiply to 3 to get 48 b
squared that is 16 b squared okay
so now now factor out nothing sha we can
now
um factor 9 a squared plus 24 a b
plus 16 b squared by it because this is
now a perfect square trinomial
okay so hapaganito i'm given a hindisha
perfect square trinomial
try to factor out and then thing
resulting from
square trinomial okay so since nothing
in a perfect square trinomial
we can have the factored form three
times
so again what is thus a square root of
9a squared that is 9a
i know 3a and then for 16b squared that
is what is the square root of 16 that is
4
and then 4b okay so the factored form is
uh 3 times the square of
the quantity of 3a plus 4b
okay let's have the next one 4x cubed
minus 24x squared plus
x so we now have since in the ul it's a
perfect square
we will factor out for x but for x that
is the gcf
e so what will you multiply to 4x to get
4x
cubed that is x squared and then to get
24x squared we will have
6x and then what will you multiply to 4x
to get 36x
that is 9. okay so check not in nine
times four that is 36 and then x
okay so we now have the perfect square
trinomial so we can now factor so
the factored form will be for x times
and a young square root
x squared that is x and on square root
99 that is 3.
so this is our factored form
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the bell button to our walmart channel
just keep on watching
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