FACTORING GENERAL TRINOMIALS || GRADE 8 MATHEMATICS Q1
Summary
TLDRThis 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
📚 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.
🔍 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.
📏 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.
Mindmap
Keywords
💡Quadratic Trinomial
💡Factoring
💡Integers
💡Product
💡Sum
💡Factored Form
💡Polynomials
💡Volume
💡Dimensions
💡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
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the bell button to our walmart channel
just keep on watching
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