Factoring Trinomials

Summary

This video introduces a unique method for factoring trinomials where the leading coefficient is not 1, a technique that may be unfamiliar to some. The presenter walks through two examples, demonstrating how to factor trinomials by multiplying the leading coefficient with the constant, finding factor pairs, and adjusting them to fit the equation. The video highlights the importance of reducing fractions and multiplying the denominators back into the equation. It's a helpful tutorial for students aiming to improve their factoring skills, especially for U.S. standardized tests.

Takeaways

• 🔢 Factoring trinomials with a leading coefficient greater than 1 can be done using a method not commonly taught in schools.
• 🧮 To start, multiply the leading coefficient by the constant value, transforming the equation.
• 🧑‍🏫 Factor the transformed trinomial by identifying two numbers that multiply to the product and add up to the middle term.
• ✅ Once the numbers are identified, create binomials for factoring.
• ➗ Divide the coefficients of the binomials by the original leading coefficient.
• 🔄 Simplify the fractions and then multiply the denominators to the variable.
• 📐 Verify the factored form by multiplying it back to the original trinomial to check for correctness.
• 🧩 In examples with negative values, use different signs in the binomials based on whether the constant is negative.
• 🔍 For subtraction cases, find pairs of numbers that subtract to give the middle term.
• 📊 Final step: multiply out the factored binomials to confirm the accuracy of the original trinomial.

Q & A

What is the first step in factoring a trinomial with a leading coefficient not equal to one?

The first step is to multiply the leading coefficient by the constant term. For example, in the expression 12x² + 17x + 6, you multiply 12 by 6, resulting in 72.

How do you factor the expression after multiplying the leading coefficient with the constant?

After multiplying, rewrite the expression as a trinomial with x². For example, 12x² + 17x + 6 becomes x² + 17x + 72. Then factor it by finding two numbers that multiply to the constant and add to the middle term.

How do you determine the two numbers when factoring the trinomial?

You list pairs of numbers that multiply to the constant (72 in the first example). Then, find the pair that adds up to the middle term's coefficient (17 in this case). For 72, the correct pair is 8 and 9 because 8 + 9 = 17.

Why do you divide the factored terms by the original leading coefficient?

You divide by the original leading coefficient (12 in this case) because earlier, you multiplied the leading coefficient to simplify the trinomial. Dividing helps balance the equation and return it to its correct form.

What do you do with the fractions after dividing the factored terms?

You reduce the fractions if possible. For example, after dividing by 12, you get terms like 2/3 and 3/4. Next, you multiply the denominators to the x terms in each factor.

How do you handle negative constants in trinomials?

When the constant is negative, like in x² - 5x - 24, you use one positive and one negative sign in the factors because the constant term determines the signs in the factor pairs.

What is the process for factoring trinomials with opposite signs in the factors?

You need to find two numbers that subtract to the middle term's coefficient. For example, to factor x² - 5x - 24, you find numbers that multiply to -24 and subtract to -5. The correct pair is 8 and -3.

How do you determine which factor gets the negative sign?

The larger number is paired with the negative sign if the middle term is negative. For instance, with -5x, you place the -8 in the factor that corresponds with x, and the +3 goes in the other factor.

What happens after you factor a trinomial with negative constants?

You divide the factored terms by the original leading coefficient, reduce the fractions, and then multiply the denominators back into the x terms.

How can you verify the factored form is correct?

You can verify the factored form by multiplying the factors back out to see if you get the original trinomial. For example, multiplying (2x + 1)(3x - 4) should give you the original 6x² - 5x - 4.

Outlines

00:00

🧠 Tackling Trinomials with a Non-1 Leading Coefficient

This paragraph introduces the common challenge of factoring trinomials where the leading coefficient is not one. It emphasizes that this method may not be familiar to everyone, but is particularly helpful for those aiming to improve their math skills, especially in the U.S. The key example starts by multiplying 12 and 6, transforming the expression into a simpler trinomial: x² + 17x + 72.

🔢 Finding Numbers that Multiply and Add Up

The next step focuses on factoring x² + 17x + 72 by identifying two numbers that multiply to 72 and add to 17. The list of possible pairs is presented, narrowing down to 8 and 9. These values are inserted into the factorized form X + 8 and X + 9, setting up the next step of the process.

➗ Dividing by the Multiplied Value

Here, the multiplication from the earlier step (12) is revisited. To balance the equation, the 8 and 9 are divided by 12, resulting in fractional expressions: X + 2/3 and X + 3/4. The paragraph sets the stage for finalizing the factorization.

✖️ Multiplying and Finalizing the Factored Form

This section describes how to simplify the factorized form by multiplying the denominators (3 and 4) to the respective X terms, resulting in 3x + 2 and 4x + 3. The process concludes by verifying that the factorized expression gives the original trinomial when expanded.

📊 Second Example: Factoring Another Trinomial

A second example is introduced, following a similar method. The trinomial starts with 6x² - 5x - 24, and the leading coefficient (6) is multiplied by the constant (-24), creating a simpler trinomial. The paragraph sets up the same factoring approach as the first example.

🔄 Handling Negative Terms in Factorization

In this step, the paragraph explains how to deal with negative terms. Since the third term is negative, one factor is positive and the other negative. The correct pair of numbers (8 and 3) is found by checking which combination subtracts to give -5. The larger number, 8, is assigned the negative sign, as it matches the -5 in the original equation.

✂️ Reducing and Finalizing the Second Example

To finalize the second example, the fraction resulting from dividing the factors by 6 is reduced, giving X + 1/2 and X - 4/3. The denominators are multiplied to their respective X terms, resulting in 2x + 1 and 3x - 4. Expanding this yields the original trinomial, confirming the factorization is correct.

📚 Conclusion and Encouragement

The final paragraph offers encouragement to the viewer, suggesting that this factoring method can be helpful, especially for trinomials with non-1 leading coefficients. The speaker thanks the audience for watching, and asks them to subscribe and give the video a thumbs up if they found it helpful.

Keywords

💡Trinomial

A trinomial is an algebraic expression consisting of three terms, typically in the form ax² + bx + c. In the video, the speaker focuses on factoring trinomials, particularly those where the leading coefficient is not 1, which requires a more complex factoring process. The examples provided involve breaking down the trinomial into two binomials.

💡Leading Coefficient

The leading coefficient is the number in front of the highest power of x in a polynomial. In this video, the leading coefficient is crucial because it is not equal to 1, which makes the factoring process more challenging. The speaker illustrates how to handle this by multiplying the leading coefficient with the constant term.

💡Factoring

Factoring refers to breaking down a polynomial into simpler components that, when multiplied, yield the original polynomial. The video teaches a method for factoring trinomials with a leading coefficient other than 1, helping students simplify and solve these expressions. This involves finding two numbers that multiply to a certain value and add to another.

💡Multiplication

Multiplication is used early in the factoring process to combine the leading coefficient with the constant term. For instance, in the first example, 12 is multiplied by 6 to get 72, which is then factored further. This step is essential for simplifying the trinomial into a factored form.

💡X-intercept

The x-intercept is where a graph crosses the x-axis, and solving a factored trinomial often leads to finding these intercepts. Although not explicitly mentioned as x-intercepts, factoring the trinomial helps identify the values of x where the expression equals zero, which are the solutions or roots of the equation.

💡Binomial

A binomial is an algebraic expression with two terms. In the video, the goal of factoring the trinomial is to express it as a product of two binomials. For example, after factoring, the trinomial is rewritten as (3x + 2)(4x + 3), where each expression is a binomial.

💡Add and Multiply

The method shown in the video hinges on finding two numbers that both multiply to a product and add to a sum. For example, in the trinomial x² + 17x + 72, the numbers 8 and 9 multiply to 72 and add up to 17, allowing the trinomial to be factored into binomials.

💡Denominator

The denominator is a key concept in the video when reducing fractions during the factoring process. For example, after dividing by 12 in one of the steps, the speaker reduces the resulting fractions and multiplies the denominators to the x in each binomial.

💡Reducing Fractions

Reducing fractions involves simplifying them by dividing the numerator and denominator by their greatest common factor. This step occurs in the factoring process when the speaker divides the resulting binomial terms by a value and simplifies the fractions before completing the factored form of the trinomial.

💡Positive and Negative Signs

The video addresses the importance of recognizing positive and negative signs when factoring trinomials, particularly when one of the terms is negative. The speaker explains how to position the numbers correctly based on their signs, ensuring the factored form reflects the original trinomial.

Highlights

Introduction to factoring trinomials where the leading coefficient is not 1.

Multiply the leading coefficient (12) by the constant term (6) to simplify the trinomial.

The expression becomes x² + 17x + 72 after multiplication.

Factor the new expression into (x + 8)(x + 9) by finding two numbers that multiply to 72 and add up to 17.

Divide the resulting factors by 12 to adjust for the original multiplication.

Simplify the fractions to get (x + 2/3)(x + 3/4).

Multiply the denominators by x to achieve the final factored form: (3x + 2)(4x + 3).

If you multiply the final factors out, you will get the original trinomial 12x² + 17x + 6.

A second example involves the trinomial 6x² - 5x - 24, starting by multiplying the leading coefficient (6) by the constant (-24).

Factor the expression into (x + 3)(x - 8), finding two numbers that multiply to 24 and subtract to 5.

Place the larger number (8) with the negative sign to match the -5x middle term.

Divide both terms by 6 to get (x + 1/2)(x - 4/3).

Multiply the denominators by x to finalize the factors as (2x + 1)(3x - 4).

Multiplying out the factors (2x + 1)(3x - 4) gives back the original trinomial 6x² - 5x - 24.

The method demonstrates a structured approach to factor trinomials with leading coefficients greater than 1.

Transcripts

we've all been there where we had to go

ahead and factor a trinomial where the

leading coefficient is not one

and here's a way that you may not have

learned in school especially if you want

to score in the U.S that's going to help

you that's very trinomial where the

leading coefficient is not one here we

go first take that 12 multiply to the

six so we'll go through 12 times 6 and

the expression becomes x squared plus

17x plus 72. next we're just going to go

ahead and Factor this to factor this we

have X Plus and X Plus and to find out

the two numbers that goes in here we

need to find two numbers that multiplies

to 72 and adds up to 17. first of all

the list of numbers that multiplies to

72 are 1 and 72 2 and 36 3 and 24 4 and

18 6 and 12 and 8 and 9. then all these

pairs of numbers that we have what two

numbers as of 17.

eight and nine so I'm going to put 8

here and I'm going to put a 9 here next

you know that 12 that we multiply where

did it go well we multiply by 12 so

we're going to divide it by

12 here then once we reduce the

fractions it becomes

X plus 2 over 3 and X plus three over

four to complete all we have to do is

take the denominator multiply to the X

take the 3 multiply to the X giving us

3x Plus 2.

next we're going to take that 4 multiply

to the X giving us 4X Plus

three and of course if you multiply this

out you'll get 12x squared plus 17x plus

6. or factoring the original expression

that would be our final answer let me do

another example

here's a second example just as in the

first example I'm going to take the

leading coefficient of 6 multiply to the

C value

where we get x squared minus 5X

minus 24. next we're going to factor

this thing and again we get the X in the

front however because this is negative

the third term it's going to be plus

minus again whatever the third term is

negative we get 1 plus and 1 minus

again we need to find two numbers that

multiplies to 24 first which are 1 and

24 to a 12 3 and 8 4 and 6. and this

time instead of adding because the two

signs that are opposite of each other we

want to find two numbers where we

subtract and get five and it's going to

be three and eight because eight minus

three of course is five only thing you

have to figure out is does the eight go

here

or the 8 goes here since that's negative

5 we want to put the bigger number with

the negative so we put the eight here

and the other number or the three goes

right there

next you'll have six that we multiply

I'm going to divide the 3 and the eight

by six and by reducing the three over

six and the eight over six we get

X plus one half and x minus four over

three

we're going to take that 2 multiply to

the X giving us 2x Plus 1. notice the

one stays the 2 is gone since we've

tacked it out to the x

again we're going to take that three

tack it on to the x or multiply where we

get 3x minus

4. again if you multiply this out you'll

get 6X squared minus 5x minus 4. or if

you factor the original expression the

trinomial once again we get 2x plus 1 3x

minus 4. so I hope this method if you

haven't seen it before will help you

back through trinomials with the leading

coefficient of this band one as always

thank you very much for watching my

videos and supporting if you haven't

subscribed please do so and give it a

thumbs up