ATHS Factoring General Trinomial

Summary

This educational video script explains the process of factoring quadratic trinomials using the 'ac method' with two cases. It demonstrates how to factor expressions like x^2 + 5x + 6 and x^2 - 7x + 12 by identifying pairs of factors that sum to the middle coefficient. The script also covers factoring general trinomials, such as 3x^2 + 13x + 4 and 4x^2 - 16x + 7, by pairing factors of the leading coefficient with those of the constant term, ensuring the inner and outer product sums match the middle term. Each example is methodically worked through to showcase the factoring technique.

Takeaways

• 🔢 The fourth type of factoring is called general factoring, which has two cases.
• 📝 The first case involves factoring trinomials like x² + 5x + 6. Start by identifying the factors of the constant (6) and find pairs that add up to the middle coefficient (5).
• 🧮 For x² + 5x + 6, the correct factor pair is (x + 3)(x + 2).
• 📘 Another example is x² - 7x + 12. The correct factor pair is (x - 3)(x - 4).
• 🔍 In the case of x² + 2x - 15, the correct factor pair is (x - 3)(x + 5).
• 🛠️ The second case of general factoring involves more complex trinomials, such as 3x² + 13x + 4. Start by identifying factors of the first and last terms.
• 📊 Position the factors in binomials, and then multiply the inner and outer terms to match the middle coefficient.
• ✔️ For 3x² + 13x + 4, the correct factor pair is (3x + 1)(x + 4).
• 🔗 Another example, 2x² - 3x - 9, factors to (2x + 3)(x - 3).
• 🔧 Finally, 4x² - 16x + 7 factors to (2x - 7)(2x - 1).

Q & A

What is the fourth type of factoring mentioned in the script?

The fourth type of factoring mentioned in the script is called 'general trinomials'.

How many cases are there for the general trinomial factoring method?

There are two cases for the general trinomial factoring method.

What is the first step in factoring a trinomial of the form x^2 + bx + c?

The first step in factoring a trinomial of the form x^2 + bx + c is to identify the factors of the constant term (c).

Why is it necessary to add the pairs of factors in the factoring process?

The pairs of factors are added to ensure that the sum of the products matches the middle coefficient (b) of the trinomial.

What is the correct pair of factors for the trinomial x^2 + 5x + 6?

The correct pair of factors for the trinomial x^2 + 5x + 6 is (x + 3) and (x + 2).

What is the sign of the factors when all the coefficients of the trinomial are positive?

When all the coefficients of the trinomial are positive, the factors will have the same signs.

How do you determine the correct pair of factors when there are multiple possibilities?

You determine the correct pair of factors by multiplying the inner and outer terms of each pair and checking if the sum matches the middle term of the trinomial.

What is the factored form of the trinomial 3x^2 + 13x + 4?

The factored form of the trinomial 3x^2 + 13x + 4 is (3x + 1)(x + 4).

What is the strategy for factoring a trinomial when the leading coefficient is not 1?

When the leading coefficient is not 1, you first factor out the greatest common factor and then apply the general trinomial factoring method to the remaining quadratic expression.

How does the script handle negative constants in the trinomial factoring process?

The script handles negative constants by considering pairs of factors that, when added, result in the negative middle term of the trinomial.

What is the significance of the middle term in the factoring process?

The middle term in the factoring process is significant because it determines the correct pair of factors when checking the sum of the products of the inner and outer terms.

Outlines

00:00

📐 Understanding General Quadratic Factoring

This paragraph introduces the fourth type of factoring, known as general quadratic factoring, which involves two cases. The focus is on factoring quadratic expressions by identifying pairs of factors of the constant term that add up to the middle coefficient. The process is demonstrated with examples: x^2 + 5x + 6 is factored into (x + 3)(x + 2), x^2 - 7x + 12 into (x - 3)(x - 4), and x^2 + 2x - 15 into (x - 3)(x + 5). Each example illustrates the method of finding factor pairs, ensuring their sum matches the middle term, and then forming the binomials.

05:00

🔍 Case Two of General Quadratic Factoring

The second case of general quadratic factoring is explored, starting with the expression 3x^2 + 13x + 4. The method involves identifying factors of the leading coefficient (3x^2) and the constant term (4), then pairing them to form binomials. The correct pair is determined by multiplying the inner and outer terms of each pair and checking if their sum equals the middle term of the quadratic (13x). The correct factorization is found to be (3x + 1)(x + 4). The process is also applied to 2x^2 - 3x - 9, resulting in the factors (2x + 3)(x - 3).

10:01

🔢 Factoring with Constant Terms and Middle Coefficients

This section continues the factoring process with examples that include identifying factors of both the leading term and the constant term, such as 4x^2 - 16x + 7. The factors of the constant term are paired with those of the leading term to form potential binomials. The correct binomials are determined by checking which pair, when multiplied, yields the middle term of the quadratic expression. For the given example, the correct factorization is found to be (2x - 7)(2x - 1), ensuring the sum of the products of the inner and outer terms matches the middle coefficient of the quadratic.

Keywords

💡factoring

Factoring is the process of breaking down a polynomial into a product of other polynomials. In the context of the video, factoring is used to simplify expressions and equations, making them easier to solve or analyze. The video focuses on different types of factoring, particularly 'general trinomials,' which are polynomials of degree two.

💡general trinomials

A general trinomial refers to a polynomial of the second degree, which has three terms. The video script discusses two cases of factoring general trinomials, emphasizing the importance of identifying the correct factors to satisfy the equation's conditions. This is central to the video's educational content on algebra.

💡constant

In algebra, the constant term is the term in a polynomial that does not contain any variables. The video script mentions focusing on the constant term when factoring, as it helps in identifying the possible factors that can be used to break down the polynomial.

💡numerical coefficient

The numerical coefficient refers to the number that multiplies a variable in an algebraic expression. In the video, the script explains that when factoring, the sum of the products of the inner and outer terms should match the middle term's numerical coefficient to ensure the factorization is correct.

💡binomials

Binomials are algebraic expressions consisting of two terms. The video script provides examples of factoring binomials, which is a fundamental algebraic skill. Binomials are often factored to simplify expressions or to find solutions to equations.

💡inner and outer terms

When factoring binomials, the terms are often grouped into 'inner' and 'outer' pairs. The video script illustrates how to multiply these terms to check if their sum matches the middle term of the original polynomial, which is a crucial step in the factoring process.

💡factor pairs

Factor pairs are two numbers that multiply together to give the product of the polynomial's constant term. The video script discusses identifying and using factor pairs to factor general trinomials, which is essential for the correct factorization of the polynomial.

💡middle term

The middle term of a polynomial is the term that falls between the highest degree term and the constant term. In the video, the script explains that the sum of the products of the inner and outer terms should equal the middle term to validate the factorization process.

💡polynomial

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The video script uses polynomials as the main objects for factoring, demonstrating various techniques to factor them.

💡degree of a polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. The video script refers to polynomials of the second degree, which are the focus of the factoring techniques being discussed.

💡quantity

In the context of the video script, 'quantity' refers to a mathematical expression that is being multiplied by another expression. The script uses the term 'quantity' to describe the factored form of polynomials, emphasizing the multiplication of binomials to achieve the original polynomial.

Highlights

Introduction to the fourth type of factoring called 'general trinomial' with two cases.

Explanation of the first case of general trinomial factoring with the example of x^2 + 5x + 6.

Identification of factors of six and the process of finding the correct pair to satisfy the middle coefficient.

Factoring of x^2 + 5x + 6 into (x + 3)(x + 2).

Second example with x^2 - 7x + 12 and the process of identifying the correct factors.

Factoring of x^2 - 7x + 12 into (x - 3)(x - 4).

Third example with x^2 + 2x - 15 and the process of finding the correct factors.

Factoring of x^2 + 2x - 15 into (x - 3)(x + 5).

Transition to the second case of general trinomials with the example of 3x^2 + 13x + 4.

Identification of factors of the first and last terms and the process of arranging them into binomials.

Multiplication of inner and outer terms to find the correct factors.

Factoring of 3x^2 + 13x + 4 into (3x + 1)(x + 4).

Second example for case two with 2x^2 - 3x - 9 and the process of identifying factors.

Factoring of 2x^2 - 3x - 9 into (2x + 3)(x - 3).

Last example for case two with 4x^2 - 16x + 7 and the process of finding the correct factors.

Factoring of 4x^2 - 16x + 7 into (2x - 7)(2x - 1).

Conclusion of the general trinomial factoring process with practical examples.

Transcripts

[Music]

the fourth type of factoring is called

general kai mom yak

it has two cases case one and place two

so we have to give an x square plus five

x plus six

since the whole equation of x squared is

one we will focus on the constant

which is six so the first step is to

identify the factors of six

what can be the factors of six

one times six and

three times two the second step is to

add

this pairs so six six plus one

is seven three plus two

is five we should satisfy the conditions

that

when the pairs are added it should

result to the middle

numerical coefficient which is 5 so this

is incorrect therefore we will use

this pair so separate the factors in

this form

so we will have two x's because it will

result to

x squared since all of the signs of the

given is positive

we will copy them and then

input three and

two therefore the factor of x squared

plus five x plus six

is the quantity of x plus three

multiplied by the quantity of x plus 2

let's have another example x squared

minus 7x

plus 12. again the first step

identify the pairs of the factors of

positive

12 so what can be the factors of

positive 12 we have

positive 6 times 2

positive 3 times 4

negative 3 times negative

four now let's add this pairs

six plus two is eight three plus four is

seven

negative three times negative four is

negative

seven but we are looking for negative

seven since

this is the middle numerical term so

therefore

we will use the pair negative three

and negative four so the factors of x

squared minus seven

x plus twelve is x

minus three and x

minus four

let's have the third example x squared

plus 2x

minus 15. so again identify the pairs

of negative 15 what can be the factors

of negative 15

we can have negative 1 times

15 or negative 3

times 5 and positive 3

times negative 5 add these factors

negative 1 plus 15 is positive 14

negative 3 plus 5 is positive 2

3 plus negative 5 is

negative 2 so we should have a result of

positive 2 therefore

we will use the factors negative three

and five so the factors of

x squared plus two x minus fifteen is

x minus three times

x plus five

[Music]

let's proceed to the case two of the

general trinomials

we have the given three x squared plus

thirteen x plus

four so the first thing to do is to

identify the factors of the first

and the last term so we have three x

squared and four

so what can be the factor of three x

squared we can have

three x times x what are the possible

factors of four

that can be two times two

and another factor can be four

times one second step is to position the

factors in two separate binomials

please feel free to rearrange the

factors so we can have

three x plus two

and x plus two

next three x

plus four and

x plus one

we can also have three x

plus one and

x plus four

the third step is to multiply the inner

terms and the

outer terms so three x

times two is six x

two times x is two x

next pair 3x times 1

is 3x 4 times

x is 4x the last pair

3x times 4 is plus

x and 1 times x

is x the fourth step

is to add the products of the inner and

outer terms

whichever matches with the middle term

of the trinomial

are the correct factors so 2x

plus 6x is 8x

therefore this is incorrect 4x

plus 3x is 7x this is also incorrect

12x plus x is 13x

therefore this is the correct pair of

four factors

so the factored form of 3x squared plus

thirteen x

plus four is three x

plus one times x plus four

let's have the second example two x

squared minus 3x

minus 9. again first identify the factor

of the first term which is 2x squared

so 2x times x

next the factors of our constant which

is negative nine

so this can be negative three

and positive three or

it can also be positive three

times negative three another factor can

be

negative nine times positive one

and negative one times positive

nine so next step is to

group the pairs

so two x minus three

and x plus three

another pair is two x plus three

and x minus three

for this factors you can have two x

minus nine and x

plus one the last pair can be

two x minus one

and x plus nine

the next step is to multiply the inner

terms and the

outer terms remember that the sum should

result

to the middle numerical coefficient

which is negative

3x so negative 3

times x is negative 3x

2x times 3 is

6x so negative 3x plus 6x is

positive 3x therefore this is

incorrect next 3

times x is 3x

2x times negative 3 is

negative 6x 3x

plus negative 6x is equal to negative 3x

which is our middle third since you

already got the correct answer

you don't have to check the other pairs

anymore

since it will be a waste of time

therefore

the factors of two x squared minus three

x minus nine

is the quantity of two

x plus three multiplied by the quantity

of

x minus three

[Music]

so let's have the last example for the

case 2 of the general binomial

so we have the given 4x squared minus

16x

plus 7. let's identify the factors of

the first term

which is 4 x squared

so you can have 4x times

x and 2x

times 2x next what can be the factor of

the constant

7 you can have positive 7

times positive 1 and negative 7

times negative one next

pair up the factors so one

factor can be four x

plus seven times x

plus one another possible factor is

four x minus seven

times x minus one

next two x plus seven

times two x

plus one

you can also have two x minus seven

times x minus one feel free to rearrange

the factor

since it is also a possible answer

next step multiply the inner terms and

the outer terms

remember that the sum should result to

the numerical coefficient of the

middle term seven times

x is seven x four x times one

is four x seven x plus

4 x is 11x therefore

this is wrong next negative 7x

4x times negative one is negative

for x negative seven x

plus negative four x is negative eleven

x which is also incorrect

next pair 7 times 2x is 14x

2x times 1 is 2x

14x plus 2x is positive

16x which is wrong

next negative 7 times 2x

is negative 14x 2x

times negative 1 is negative 2x

negative 14x plus negative 2x

is negative 16x which is our

middle term therefore the factory

or the factors of 4x squared minus 16x

plus 7 is the quantity of 2x minus 7

multiplied by 2x

minus 1.

[Music]

foreign