Factoring Sums and Differences of Perfect Cubes
Summary
TLDRThis educational video delves into the method of factoring expressions involving sums and differences of cubes. It introduces the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2) for sums and a^3 - b^3 = (a - b)(a^2 + ab + b^2) for differences, using a and b as cube roots of the terms. Through examples like factoring x^3 + 8 and x^3 - 216, the video simplifies complex algebraic expressions, making the process accessible and emphasizing the importance of recognizing perfect cubes for effective factoring.
Takeaways
- 🔢 The formula for factoring a sum of cubes is ( a^3 + b^3 = (a + b)(a^2 - ab + b^2) ).
- 🔑 In the sum of cubes formula, a and b are the cube roots of the terms being factored.
- 📚 For example, to factor x^3 + 8, identify a = x and b = 2 (since 2^3 = 8) and apply the formula.
- ✅ Practice by factoring expressions like x^3 + 125 and 8x^3 + 27 using the sum of cubes formula.
- 🔄 The formula for factoring a difference of cubes is ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ).
- 📉 For a difference of cubes, the sign between a and b in the factored form is negative.
- 📘 Examples of difference of cubes include factoring x^3 - 216 and 64y^3 - 125.
- 🔄 The sign in the factored form of a sum or difference of cubes flips from the first term to the last.
- 💡 The script provides a generalized formula for both sum and difference of cubes, highlighting the sign changes.
- 📌 The script emphasizes the importance of recognizing perfect cubes and applying the formulas correctly.
Q & A
What is the main focus of the video?
-The main focus of the video is to teach factoring sums and differences of cubes.
What is the formula for factoring a sum of cubes?
-The formula for factoring a sum of cubes is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).
How do you determine the values of 'a' and 'b' in the sum of cubes formula?
-In the sum of cubes formula, 'a' is the cube root of the first term and 'b' is the cube root of the second term.
What is an example of factoring a sum of cubes given in the video?
-An example given in the video is factoring \( x^3 + 8 \), where 'a' is \( x \) and 'b' is \( 2 \), resulting in \( (x + 2)(x^2 - 2x + 4) \).
What is the formula for factoring a difference of cubes?
-The formula for factoring a difference of cubes is \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).
How do you factor the expression \( x^3 + 125 \) using the sum of cubes formula?
-For the expression \( x^3 + 125 \), 'a' is \( x \) and 'b' is \( 5 \), leading to the factored form \( (x + 5)(x^2 - 5x + 25) \).
What is the process for factoring the expression \( 8x^3 + 27 \)?
-For \( 8x^3 + 27 \), 'a' is \( 2x \) and 'b' is \( 3 \), resulting in the factored form \( (2x + 3)(4x^2 - 6x + 9) \).
How does the video handle the factoring of an expression that is not a perfect cube?
-The video suggests replacing non-perfect cubes with the closest perfect cube and then proceeding with the factoring process.
What is the generalized formula for both sum and difference of cubes?
-The generalized formula for both sum and difference of cubes is \( a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) \).
Can you provide an example of factoring a difference of cubes from the video?
-An example from the video is factoring \( x^3 - 216 \), where 'a' is \( x \) and 'b' is \( 6 \), resulting in \( (x - 6)(x^2 + 6x + 36) \).
How does the video explain the sign change in the generalized formula for factoring cubes?
-The video explains that in the generalized formula, the sign before 'ab' changes from plus to minus for the difference of cubes and remains the same for the sum of cubes.
Outlines
📚 Factoring Sums and Differences of Cubes
This paragraph introduces the concept of factoring expressions involving sums and differences of cubes. The presenter explains the formula for factoring a sum of cubes, a^3 + b^3, which is (a + b)(a^2 - ab + b^2). Using the example x^3 + 8, the presenter demonstrates how to equate a^3 with x^3 and b^3 with 8, and then substitute the values into the formula to factor the expression. The process is repeated with the expression x^3 + 125, where the cube root of 125 is identified as 5, and the formula is applied to factor the expression. The paragraph also covers the factoring of a sum of cubes with variables and different coefficients, such as 8x^3 + 27, and 25x^3 + 64y^3, emphasizing the importance of identifying the correct values for a and b before applying the formula.
🔢 Difference of Cubes and General Formula
The second paragraph delves into the factoring of the difference of cubes, using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). The presenter clarifies that when dealing with a difference of cubes, the sign between a and b in the factored form is negative, and the last term in the formula changes to a positive sign. Examples are provided to illustrate the process, such as factoring x^3 - 216, where the cube root of 216 is 6, and the formula is applied accordingly. The paragraph also covers the factoring of 64y^3 - 125 and 8y^3 - 27, showing how to identify a and b, and then correctly apply the formula. Towards the end, the presenter introduces a generalized formula for both sums and differences of cubes, highlighting the change in sign between the terms.
🧮 Advanced Factoring with Variables and Cube Roots
The final paragraph presents more complex examples involving the factoring of expressions with higher powers and multiple variables. The presenter begins with the expression x^6 - 64y^3, explaining how to find the cube roots of x^6 and 64y^3 to identify a and b. The formula for the difference of cubes is then applied to factor the expression. The paragraph demonstrates the process of simplifying the terms and correctly applying the formula to obtain the factored form. The presenter also addresses the importance of understanding the cube roots and the correct application of the formula for both sums and differences of cubes, providing a comprehensive understanding of the factoring process.
Mindmap
Keywords
💡Factoring
💡Sums and Differences of Cubes
💡Cube Root
💡Algebraic Expressions
💡Formula
💡Cube
💡Factor
💡Expression
💡Equation
💡Simplify
Highlights
Introduction to factoring sums and differences of cubes.
Explanation of the formula for factoring a sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Example of factoring X^3 + 8 using the sum of cubes formula.
Identification of a and b in the formula for the given expression X^3 + 8.
Substitution of a and b values into the formula to factor the expression.
Encouragement for the viewer to pause and try factoring X^3 + 125.
Factoring of X^3 + 125 using the sum of cubes formula.
Example of factoring 8x^3 + 27 using the sum of cubes formula.
Introduction to factoring the sum of cubes with variables: 25x^3 + 64y^3.
Correction of the example to use 27 instead of 25 for factoring.
Factoring of 27x^3 + 64y^3 using the sum of cubes formula.
Introduction to the formula for factoring a difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Example of factoring X^3 - 216 using the difference of cubes formula.
Factoring of 64y^3 - 125 using the difference of cubes formula.
Example of factoring 8y^3 - 27 using the difference of cubes formula.
Generalized formula for factoring both sums and differences of cubes.
Example of factoring X^6 - 64y^3 using the generalized formula.
Conclusion on the simplicity of factoring cubes with the provided formulas.
Transcripts
in this video we're going to focus on
factoring sums and difference of Cubes
so let's say if we want to factor the
expression X Cub +
8 now there is an equation that you want
to use and here it
is a the 3 + b the
3 this is equal to a +
b time uh a 2
minus
AB plus
b^2 so you need to realize that a to the
3 is the same as X cub in this problem
therefore if you take the cube root of
both sides a is equal to X now B the 3
is equal to 8 and the cube root of 8 is
two so B is equal to two and now we just
got to plug in everything into the
formula so just keep in mind a is X B is
2 so A + B that's going to be x +
2 a 2 is x^2 a * b or x * 2 that's
2X and b^2 is 2^ 2 2 * 2 is 4 and so
that's how you can Factor uh this
particular expression but let's go ahead
and try another
example let's say if we want to factor
the
expression X Cub +
125 feel free to pause the video and try
this
example so notice that a to the
3 is the same as X Cub for this problem
therefore a is
X and B the 3 is
125 now what is the cube root of 125
what times what * 1 is
125 the answer is
five so instead of writing a plus b this
is going to be x +
5 and then it's multiplied by A2 or x^2
minus a * b or 5 * X Plus b^2 which is
5^
2 and so that's how you can factor a sum
of
cubes now let's try and another example
let's
say if it's 8 x 3r +
27 try that
example so we can see that a the 3 is
equal to 8 x Cub so if that's the
case what is the value of
a the cube root of 8 is 2 and the cube
root of x 3r is simply X so a is
2x now B Cub is
27 and we need to take the cube root of
27 to find B that means B is equal to 3
so now using the formula A + B it's
going to be 2x + 3 * a^ 2 which is 2x^2
or 2x * 2X and that's
4x^2 minus a which is 2 2 x * 3 and
that's 6 x + b^ 2 which is 3^ 2 or 9 so
that's the
answer so let's try one more example
with sum of
cubes try this one 25 x Cub + 64
YB so we can see that a cub is 25x cub
and B
Cub is 64 y
Cub actually I can't use 25 let's take
out
25 and let's use uh 27
instead 25 is not a perfect
Cube so a is going to be the cube root
of 27 which is three and the cube root
of x Cub is X B is going to be the cube
root of 64
and the cubot of Y cube is y so now that
we have a and b we could find the
answers so now let's
substitute a is
3x and B is 4 Y and then it's a 2 or 3x^
2us a B which is uh 3x * 4 y
+ B ^2 or 4 y^
2 so this is equal to 3x + 4
y now 3x^2 that's 3x * 3x which is going
to be 9
x^2 -3x * 4 y it's going to
bexy and 4 y^2 4^ 2 is 16 so this is
going to be plus 16
y^2 so this is the
answer now the next equation that you
need to be familiar with is the
difference of Cubes so a cub minus BB
and this is equal to a minus B * a 2 +
AB plus b
s so if there's a negative sign
between a cub and B Cub there going to
be a negative sign between a and b and
then this sign is going to change to a
positive in the last example we had this
equation A 3r + B 3r is equal to a + b *
a 2 minus a + b
2 so if you want to come up with a a
generalized formula for both here it is
this is going to be plus and then
minus actually let me put it in
different
colors so it's going to be a plus or
minus and then a
s the sign is going to flip at this
point so I'm going to put the the blue
one on top but it's going to be minus
the red one on the bottom that's going
to be plus
AB +
B2 so as you can
see the sign is going to stay the same
at first and then it's going to
reverse now let's try some
examples try this one X Cub -
216 so a is equal to X and B is the cube
root of 216 which is 6
so this is going to be x - 6 * a^ 2 or
x^2 + a or + 6 * x + b^2 which is 6^ 2
or
36 and that's it for that example let's
try some
more try this
one
64 y
Cub
minus 125
so a cub is 64
YB B 3
is25 the cube root of 125 is 5 the cube
root of 64 is four so a is going to be 4
y so a minus B that's 4 y - 5 * a^ 2
which is 4 y^ 2 or 4 y * 4 Y which is 16
y^ 2 plus a or 4 y * 5 which is uh 20 y
+ b^ 2 or 5^ 2 which is
25 try this one 8 y cubus
27 so a is or a cub is 8 y Cub B to the
3 is 27 7 so a is going to be 2 Y and B
is equal to
3 so it's going to be a - b or 2 y - 3 *
a^ 2 which is 2 y * 2 Y and that's 4 y^
2+ a 2 y * 3 is 6 y plus b^ 2 or 3^ 2
which is n so as you can
see these problems are not too difficult
to do
but let's try some different examples
try this
one x
6 -
64 y
9th so a
cub is X to the 6 power and B Cub is 64
y
9th so what is the cube root of
x 6 to find a cube root you can raise
both sides to the 1/3 power so basically
you're dividing 6 by 3 6id 3 is 2 so a
is
X2 now what about B the cube root of 64
is 4 the cube root of Y the 9 is
basically 9 / 3 so it's going to be y
the
3r so a minus B that's x^2 - 4 y
Cub now what's a
s so if a is x s a 2 is going to be X to
4th now what about
AB so multiply X2 and 4 y Cub that's
going to be
positive 4 x^2 y Cub plus b^2 now B is 4
YB so b^ 2 is 4 * 4 which is 16 and YB *
YB is y
6 so it's + 16 y 6 that's the answer
関連動画をさらに表示
Solving Quadratic Equations using Quadratic Formula
Factoring Binomials & Trinomials - Special Cases
Evaluating Functions - Basic Introduction | Algebra
FACTORING GENERAL TRINOMIALS || GRADE 8 MATHEMATICS Q1
Average vs Instantaneous Rates of Change
Factoring Part 1 - Common Monomial Factoring | Grade 8 Q1 @MathTeacherGon
5.0 / 5 (0 votes)