Factoring Trinomials
Summary
TLDRThis 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
🧠 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.
Mindmap
Keywords
💡Trinomial
💡Leading Coefficient
💡Factoring
💡Multiplication
💡X-intercept
💡Binomial
💡Add and Multiply
💡Denominator
💡Reducing Fractions
💡Positive and Negative Signs
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
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