FACTORING PERFECT SQUARE TRINOMIALS || GRADE 8 MATHEMATICS Q1

Summary

This 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

00:00

📚 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.

05:02

🔍 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.

10:04

📐 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.

15:07

📘 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.

Keywords

💡Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. It consists of three terms: the first term is the square of a number or variable, the middle term is twice the product of the square roots of the first and third terms, and the third term is the square of another number or variable. In the video, this concept is central as it is used to determine which expressions can be factored into a squared binomial and which cannot.

💡Factor

To factor in mathematics means to express a polynomial as the product of its factors. In the context of the video, factoring a perfect square trinomial involves rewriting it as the square of a binomial. For example, the script mentions factoring expressions like 'x squared plus 2xy plus y squared' into '(x + y) squared'.

💡Middle Term

In the context of a trinomial, the middle term is the second term in a three-term polynomial. The video explains that for a trinomial to be a perfect square, the middle term must be twice the product of the square roots of the first and third terms. For instance, in '4x squared plus 20x plus 25', the middle term '20x' is twice the product of the square roots of '4x' and '5'.

💡First Term

The first term of a trinomial is the initial term in a three-term polynomial. In the video, it is emphasized that for an expression to be a perfect square trinomial, the first term must be a perfect square and positive. An example given is 'x squared', which is a perfect square of 'x'.

💡Last Term

The last term in a trinomial is the final term of the polynomial. The video explains that, similar to the first term, the last term in a perfect square trinomial must also be a perfect square and positive. An example used in the script is 'y squared', which is the square of 'y'.

💡Positive

In mathematics, a positive number is any value greater than zero. The video script specifies that both the first and last terms of a perfect square trinomial must be positive perfect squares. This is illustrated with terms like '4x squared' and '25', which are both positive and perfect squares.

💡Binomial

A binomial is an algebraic expression with two terms. In the context of perfect square trinomials, the video discusses expressing these trinomials as the square of a binomial, such as '(x + y)', when the trinomial is a perfect square.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The video uses the concept of square roots to identify the terms of a perfect square trinomial and to factor such expressions. For example, the square root of '16' is '4', as 4 times 4 equals 16.

💡Product

In mathematics, the product is the result of multiplying two or more numbers together. The video explains that the middle term of a perfect square trinomial is twice the product of the square roots of the first and last terms. An example from the script is '2 times 2x times 5', which equals '20x'.

💡Greatest Common Factor (GCF)

The GCF of two or more numbers is the largest number that divides evenly into each of them. In the video, the GCF is used to simplify expressions before determining if they are perfect square trinomials or to factor expressions. For example, '4x cubed' has an GCF of 'x', which can be factored out to simplify the expression.

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

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