Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 7
Classify each binomial as either a sum of cubes, a difference of cubes, a difference of squares, or none of these. $$ 25 x^{2}+8 x $$
Short Answer
Expert verified
None of these
Step by step solution
01
Identify terms
Examine the given binomial $$25x^{2}+8x$ to identify if it fits the form of any special binomial patterns. These patterns include the sum of cubes, difference of cubes, and difference of squares.
02
Check the pattern for a Sum of Cubes
The sum of cubes follows the pattern: $$a^3 + b^3.$$ Examine the exponents: The given binomial has $$25x^{2}$$ and $$8x$$. The cube would require an exponent of $$3.$$ Since there is no variable raised to the power of $$3$$ in the binomial, it cannot be expressed as a sum of cubes.
03
Check the pattern for a Difference of Cubes
The difference of cubes follows the pattern: $$a^3 - b^3.$$ Again, examine the exponents: $$25x^{2}$$ and $$8x.$$ The cube would require an exponent of $$3.$$ Since there is no variable with an exponent of 3 in the binomial, it cannot be expressed as a difference of cubes.
04
Check the pattern for a Difference of Squares
The difference of squares follows the pattern: $$a^2 - b^2.$$ Examine the given binomial: $$25x^{2}+8x.$$ The exponents in a difference of squares pattern must be $$2.$$ Additionally, the terms must be subtracted. Since $$25x^{2}$$ has the exponent of $$2$$ but the terms are not subtracted (they are added instead), it cannot be expressed as a difference of squares.
05
Conclusion
Comparing the given binomial $$25x^{2}+8x$$ to recognized patterns of special binomial forms, it does not match any of them. This implies that the binomial is none of these special forms.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
A difference of squares is a specific type of binomial that fits the pattern: \[a^2 - b^2\] This pattern can be factored into: \[(a - b)(a + b)\] The key thing to note here is the presence of subtraction and that both terms must be perfect squares. For instance, in the binomial \[25x^2 - 16\] , both \[25x^2\] (which is \((5x)^2\)) and \[16\] (which is \(4^2\)) are perfect squares and they are subtracted. This means it is a classic difference of squares which factors to: \[(5x - 4)(5x + 4)\]. Remember:
- The terms must be perfect squares.
- The operation must be subtraction between them.
Sum of Cubes
When dealing with sums of cubes, the pattern you need to recognize is: \[a^3 + b^3\] Cubic terms involve exponents of 3. This pattern can be factored using: \[(a + b)(a^2 - ab + b^2)\] For example, \[27x^3 + 8\] is a sum of cubes because \[27x^3\] (which is \[(3x)^3\]) and \[8\] (which is \[2^3\]) are both cubes. It factors to: \[(3x + 2)(9x^2 - 6x + 4)\]. Key points for recognizing sums of cubes:
- The terms must have an exponent of 3.
- The operation must be addition between them.
Difference of Cubes
A difference of cubes follows a specific pattern similar to the sum of cubes but involves subtraction: \[a^3 - b^3\] This can be factored as: \[(a - b)(a^2 + ab + b^2)\] Take for instance, \[64x^3 - 27\], where \[64x^3\] (which is \[(4x)^3\]) and \[27\] (which is \[3^3\]) are both cubes, and they are subtracted, thus creating a difference of cubes. This difference of cubes can be factored into: \[(4x - 3)(16x^2 + 12x + 9)\]. To identify a difference of cubes, ensure:
- Both terms have an exponent of 3.
- Subtraction is between the terms.
Binomial Patterns
Recognizing binomial patterns is crucial in algebra for simplifying expressions. Common patterns include:
- Difference of Squares: \(a^2 - b^2\)
- Sum of Cubes: \(a^3 + b^3\)
- Difference of Cubes: \(a^3 - b^3\)
- Ensure the terms are perfect squares or cubes.
- Check if they adhere to subtraction or addition.
- Compare with the standard forms.
