FACTORING GENERAL TRINOMIALS || GRADE 8 MATHEMATICS Q1

Summary

This educational video script focuses on factoring quadratic trinomials of the form x^2 + bx + c and solving related polynomial problems. It guides viewers through listing integer factor pairs for given numbers, identifying pairs that match the middle term 'b' and product 'c', and then forming the factored expression (x + m)(x + n). Examples are provided to demonstrate the process, including cases where trinomials cannot be factored with integer coefficients, highlighting the concept of prime trinomials. The script concludes with a practical application, using factoring to determine the dimensions of a box from a given volume expression, emphasizing the importance of understanding both positive and negative integer pairs in factoring.

Takeaways

• 📘 The video focuses on factoring quadratic trinomials of the form \(x^2 + bx + c\) and solving problems involving factors of polynomials.
• 🔢 It begins by listing all pairs of integers or factors for given numbers, which is a preparatory step for factoring trinomials.
• 📐 The process involves identifying pairs of integers whose product equals \(c\) (the constant term) and whose sum equals \(b\) (the coefficient of the middle term).
• 🔑 The factored form of a quadratic trinomial \(x^2 + bx + c\) is presented as \((x + m)(x + n)\), where \(m\) and \(n\) are the chosen integers.
• 🌰 Several examples are provided to demonstrate the factoring process, including \(x^2 + 10x + 16\), \(x^2 - 9x + 18\), and \(x^2 - 2x - 24\).
• ❌ The video clarifies that not all quadratic trinomials can be factored with integer coefficients, such as \(x^2 + 3x + 3\), which is a prime trinomial.
• 📦 An application of factoring is shown by solving for the dimensions of a box given its volume represented by a cubic polynomial expression.
• ✂️ The video demonstrates that if the polynomial is not a quadratic trinomial, such as \(4x^3 + 16x^2 - 48x\), it should first be simplified by factoring out the greatest common factor.
• 📉 The importance of considering both positive and negative integer pairs in the factoring process is emphasized, with examples illustrating how to choose the correct pair based on the sum and product requirements.
• 🎓 The video concludes with a summary of the key points and a call to action for viewers to engage with the content by liking, subscribing, and watching more.

Q & A

What is the main topic of the video?

The main topic of the video is factoring quadratic trinomials of the form x^2 + bx + c and solving problems involving factors of polynomials.

How does the video begin?

The video begins by listing all pairs of integers or factors for various numbers such as 4, 6, 8, 10, 12, 18, 20, and 30.

What is the general form of a quadratic trinomial?

The general form of a quadratic trinomial is x^2 + bx + c.

What is the factored form of a quadratic trinomial?

The factored form of a quadratic trinomial is (x + m)(x + n), where b is the sum of m and n, and c is their product.

How does the video demonstrate the process of factoring a quadratic trinomial?

The video demonstrates the process by first listing pairs of integers whose product is equal to 'c' and then choosing a pair whose sum is equal to 'b' to form the factors (x + m) and (x + n).

What is the first example given in the video for factoring a quadratic trinomial?

The first example given is x^2 + 10x + 16, where the factors are identified as (x + 2) and (x + 8).

What is the significance of choosing a pair of integers whose sum is equal to 'b'?

Choosing a pair of integers whose sum is equal to 'b' ensures that when the factors (x + m) and (x + n) are expanded, the middle term of the trinomial matches the original expression.

What does the video say about quadratic trinomials that cannot be factored using integer coefficients?

The video states that if a quadratic trinomial cannot be factored using integer coefficients, it is an example of a prime trinomial.

How does the video handle a polynomial with a degree higher than two?

The video first factors out the greatest common factor (GCF) for polynomials with a degree higher than two, reducing the problem to factoring a quadratic trinomial.

What is the application of factoring quadratic trinomials shown in the video?

The video shows an application of factoring quadratic trinomials by solving a problem to find the dimensions of a box given its volume represented by a cubic polynomial.

What is the final advice given by the video to viewers?

The video advises viewers to ensure that the integers chosen for factoring are either both positive or both negative, depending on the sign of the middle term 'b'.

Outlines

00:00

📚 Introduction to Factoring Quadratic Trinomials

This paragraph introduces the concept of factoring quadratic trinomials of the form x^2 + bx + c. The video script begins with an exercise to list all pairs of integers or factors for given numbers, which serves as a foundation for understanding how to factor quadratic trinomials. The process involves identifying pairs of integers whose product equals the constant term 'c' and whose sum equals the middle term 'b'. The factored form of the quadratic trinomial is then presented as the product of (x + m) and (x + n), where 'b' is the sum and 'c' is the product of the chosen pair of integers. An example is given to demonstrate the process of factoring x^2 + 10x + 16 by finding the appropriate pairs of integers.

05:01

🔍 Factoring with Negative Middle Terms

The second paragraph delves into factoring quadratic trinomials with negative middle terms. It explains that when the middle term is negative, the pairs of integers must also be considered with negative values. The script lists pairs of integers for the product of 18 and demonstrates the process of selecting a pair whose sum equals the negative middle term. The factored form of x^2 - 9x + 18 is then derived as (x - 3)(x - 6). Another example with x^2 - 2x - 24 is presented, showing that the larger integer must be negative, and the factored form is (x + 4)(x - 6). The paragraph also includes an example of a trinomial that cannot be factored with integer coefficients, x^2 + 3x + 3, highlighting the concept of prime trinomials.

10:01

📏 Applying Factoring to Solve Real-World Problems

The final paragraph demonstrates the application of factoring to solve a real-world problem involving the dimensions of a box. The given expression 4x^3 + 16x^2 - 48x is not a quadratic trinomial due to its highest degree being three. The script shows how to factor out the common term '4x', resulting in 4x(x^2 + 4x - 12). It then proceeds to factor the quadratic trinomial inside the parentheses, identifying the pairs of integers for the product of -12 and choosing the pair that sums up to 4. The dimensions of the box are then given as 4x, x + 6, and x - 2. The paragraph concludes with a summary of the key points discussed in the video and a call to action for viewers to engage with the content.

Keywords

💡Quadratic Trinomial

A quadratic trinomial is a polynomial of degree two with three terms. In the context of the video, quadratic trinomials are of the form x^2 + bx + c, where x is the variable, and b and c are constants. The video focuses on factoring these trinomials, which involves breaking down the polynomial into the product of binomials. An example from the script is 'x squared plus 10x plus 16', which is a quadratic trinomial.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of simpler polynomials or factors. The video script describes factoring as a method to simplify quadratic trinomials by finding two integers whose product is equal to 'c' and whose sum is equal to 'b'. This is illustrated in the script with the example 'x squared plus 10x plus 16' being factored into '(x + 2)(x + 8)'.

💡Integers

Integers are whole numbers that can be positive, negative, or zero. In the video, integers are used to find pairs that will help in factoring quadratic trinomials. The script lists pairs of integers whose product equals the constant term 'c' of the trinomial, and then selects a pair that sums to the coefficient 'b'. For instance, the integers 2 and 8 are used to factor 'x^2 + 10x + 16'.

💡Product

In the context of the video, the product refers to the result of multiplying two numbers. When factoring quadratic trinomials, the product is used to determine the constant term 'c'. The video script instructs viewers to list pairs of integers whose product equals 'c', as seen when factoring 'x^2 + 10x + 16' where the product of 2 and 8 equals 16.

💡Sum

The sum in the video refers to the total obtained by adding two or more numbers. In factoring quadratic trinomials, the sum is used to determine the middle coefficient 'b'. The script demonstrates this by choosing integer pairs whose sum equals 'b', such as selecting 2 and 8 because their sum is 10, which matches 'b' in 'x^2 + 10x + 16'.

💡Factored Form

The factored form is the representation of a polynomial as the product of its factors. The video explains that the factored form of a quadratic trinomial is '(x + m)(x + n)', where 'm' and 'n' are the integers found through the factoring process. An example from the script is the trinomial 'x^2 + 10x + 16' which is expressed in factored form as '(x + 2)(x + 8)'.

💡Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. The video discusses factoring polynomials, specifically quadratic trinomials. The script uses the term to introduce the concept of factoring and to explain how to break down polynomials into simpler expressions.

💡Volume

Volume in the video refers to the amount of space occupied by a three-dimensional object. The script uses the concept of volume to solve a problem involving a box with a given volume represented by the polynomial '4x^3 + 16x^2 - 48x'. The video demonstrates how to factor this polynomial to find the dimensions of the box.

💡Dimensions

Dimensions are the measurable extents of an object in space, typically length, width, and height. In the video, the script uses the concept of dimensions to relate the factoring of polynomials to real-world applications, such as finding the dimensions of a box given its volume. The factoring process helps in determining the linear factors that represent the dimensions.

💡Prime Trinomial

A prime trinomial is a polynomial that cannot be factored over the integers. The video mentions prime trinomials in the context of quadratic trinomials that do not yield integer factors. An example given in the script is 'x^2 + 3x + 3', which cannot be factored using integer coefficients, thus it is a prime trinomial.

Highlights

Introduction to factoring quadratic trinomials of the form x^2 + bx + c.

Listing all pairs of integers or factors for given numbers like 4, 6, 8, 10, 12, 18, 20, 30.

Explanation of how to factor a quadratic trinomial by finding pairs of integers whose product is 'c' and sum is 'b'.

Factored form of a quadratic trinomial is presented as (x + m)(x + n) where b is the sum and c is the product.

Example of factoring x^2 + 10x + 16 by identifying the correct pair of integers (2, 8).

Demonstration of factoring x^2 - 9x + 18 using negative integers pairs (-3, -6).

Case study on x^2 - 2x - 24, illustrating the selection of integer pairs (4, -6) for factoring.

Factoring x^2 + 3x - 10, choosing the pair (-2, 5) to achieve the sum of 'b'.

Discussion on x^2 + 3x + 3, explaining why it cannot be factored with integer coefficients.

Introduction of a problem-solving approach using factoring to find dimensions of a box with a given volume expression.

Factoring out 'x' from the cubic expression 4x^3 + 16x^2 - 48x to simplify the problem.

Factoring the resulting quadratic trinomial x^2 + 4x - 12 into (x + 6)(x - 2).

Conclusion on the importance of choosing integer pairs that are either both positive or both negative.

Advice on how to handle cases with negative integers in the factoring process.

Final thoughts and a call to action for viewers to like, subscribe, and watch more videos on the channel.

Transcripts

[Music]

in this video

we will factor quadratic trinomials of

the form

x squared plus bx plus c

also we will solve problems involving

factors of polynomials all right

let's start so first let us try to list

all pairs of integers or factors of the

following numbers

so ibignatin lahatnam possible pairs

of integers or factors numbers

so first we have four so we have one and

four or

two and two okay and then six we have

one and six two and three

for eight we have one and eight two and

four

for ten we have one and ten two and five

let's have another so we have twelve so

we have

one and twelve two and six three and

four

for eighteen we have one and eighteen

two and nine three and six for twenty we

have

one and twenty two and ten four and five

for thirty we have one and thirty

2 and 15 3 and 10 so these are just

the x these are just examples okay

[Music]

in factoring general trinomials okay

so a quadratic trinomial is just part

of or an example of quadratic general

trinomials

okay so how to factor quadratic

trinomial

quadratic okay so since x squared plus b

x plus c this is an example of quadratic

trinomial

so panu natin sha if a factor first we

will list all pairs of

integers whose product is

c pairs of integers

middle term which is your b okay so

therefore the factored form of quadratic

trinomial

x squared plus b x plus c is equal to

the product of x plus m and x plus

n all right so

liteco x squared plus b x plus c

this is our factored form x plus m times

x

plus n where b is the sum

and c is the product okay so you will

think of two integers pairs of

integers

[Music]

the resulting product must be c and this

is our

factored form let's have an example

i have x squared plus 10 x plus 16.

so in the given example first list all

pairs of integers whose product is

c all right so tatanda and

bb gate item pairs of integers data type

eating

all right so we will have what are the

factors of sixteen

we have one and sixteen two and eight

four and four okay so nistana

integers pairs of integers next we will

choose a pair whose sum

is b so pili kadito nakapag inadmo

so obviously it's two and eight

okay so we will use two and eight so

therefore

this will serve as our m and

n remember our factor is in the form of

x plus m and times x plus n

so therefore the factored form of x

squared plus 10

x plus 16 is equal to the product of

x plus two times x plus eight next

so i have x squared minus nine x plus

eighteen so again lista natin lahat

will be positive now since negative

middle term not n therefore

the patient integer is nothing but a

negative

so we will have now negative 1 and

negative 18

negative 2 and negative 9 negative 3 and

negative 6. so all of these

uh pairs of integers are all products

factors of 18 okay

so next we will choose a pair whose sum

is b so that path negative nine so

alindito

ang negative nine kapaginat nathan

so we have negative three and negative

six and that is negative nine

therefore our factored form

is x uh from x squared minus nine x plus

eighteen is

x minus three and x minus

six

next i have x squared minus two x minus

24. so this is a different case

okay so first list all the pairs of

integers so again

the total thing in since negative ito

big sub pairs of integers

so the larger integer must be negative

integer you know positive not n s

okay so therefore we have 1 and negative

24

2 and 12 negative 12 4 and negative 6

3 and negative 8. so all of these are

factors of

24. now since negative nito and larger

integer not in a negative so

obviously you ate nothing you know

negative young six nothing you know

negative

same as 12 and 24 all right

so next we will choose a pair whose sum

is b

negative this one

so we have 4 and negative 6 and that is

equal to

negative 2 so therefore our factored

form

of x squared minus 2x minus 24

is equal to the product of x plus 4 and

x minus 6.

next i have x squared plus 3x minus 10.

so first step

same procedure you have to list all the

pairs of

integers whose product is c

okay

larger integer so your smaller integer

union negative

all right so we have negative one and

ten

negative two and five so

okay so next choose a pair whose sum is

b

so pd

a positive three obviously it's negative

2

and 5 so that is 3. so therefore our

factored form

a of x squared plus 3x minus 10

is equal to x minus 2 and x

plus five

next i have x squared plus three x plus

three

so again list all the pairs of integers

whose product

is c so since

so we have

one and three now since

one and three is equal to

since we do not have a choice we only

have these

factors so one plus three is equal to

four

it doesn't satisfy the the quadratic

trinomial

so then x squared plus three x plus

three

cannot be factored using integer

coefficients

then it is an example of

prime trinomials nothing prime trinomial

because it cannot be factored

next let us try to solve a problem

so use factoring to find the dimensions

of the given box

with volume represented by the

expression

4x cubed plus 16x squared minus 48x

that the given expression or trinomial

is not a quadratic trinomial

why because the highest degree is

three so alumni that the quadratic

trinomial

must be on the second degree

so this is three so this is not a

quadratic

trinomial so first we will factor out

for x so we will have

four x times x squared plus four x minus

twelve

okay so etherness

for x this will be the result

all right so now we have now

this quadratic trinomial so we can now

factor

okay so factor x squared plus 4x minus

12

so we will just copy for x and then x

squared plus 4x minus 12. so again

since negative toda but on the long

integers nothing it's

is some positive is a negative

this is the resulting sum okay so 6

and negative 2 is equal to 4

6 times negative 2 that is negative 12

all right so therefore the dimension

of the box are 4x x plus 6

and x minus 2.

all right let's wrap up so again

it's either both positive or both

negative lanyan

okay so pano natin malalaman

next so since positive volt it's either

both positive or both negative so

since negative integers

negative next i have

uh minos

a positive is a negative how

and then the last case if this is

negative

again you must have one positive and one

negative integer

sodium hating and eviksabihen that your

larger integer must be

negative

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