Recognize that the expression is a sum of cubes, which can be factored using the formula for the sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)Here, 8a^3 can be written as (2a)^3 and 1 as 1^3.
02
Apply the sum of cubes formula
Using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2), identify a = 2a and b = 1. Substitute these into the formula: (2a)^3 + (1)^3 = (2a + 1)((2a)^2 - (2a)(1) + (1)^2)
03
Simplify the factors
Expand and simplify the terms inside the parentheses: (2a)^2 = 4a^2(2a)(1) = 2a(1)^2 = 1Therefore, the expression becomes: (2a + 1)(4a^2 - 2a + 1)
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Cubes
The sum of cubes is a specific type of expression in algebra. It looks like this: a^3 + b^3. In simpler terms, it involves adding the cubes of two numbers. It's helpful to recognize this form because it has a straightforward factoring formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Understanding this pattern will make it easier to factor algebraic expressions effectively. Identifying the sum of cubes early on can save you time and effort in solving polynomial equations.
For instance, if you have 8a^3 + 1, break it down into cubes: 8a^3 is (2a)^3 and 1 is 1^3. Once identified, you can apply the formula.
Polynomial Factoring
Factoring polynomials is a crucial skill in algebra. It involves breaking down a polynomial into simpler 'factor' polynomials whose product equals the original polynomial. Polynomials come in various forms, like a sum of cubes, which we see with 8a^3 + 1.
When factoring polynomials, always look for patterns and identities, such as the sum of cubes formula. This helps simplify expressions and solve equations more easily. In the example of 8a^3 + 1, we first recognized it as a sum of cubes and then used the formula: \( (a + b)(a^2 - ab + b^2) \). This results in (2a + 1)(4a^2 - 2a + 1).
Remember: Practice makes perfect! The more you practice identifying and factoring different forms, the more skilled you become.
Algebraic Identities
Algebraic identities are powerful tools in algebra. They provide standard forms to recognize and simplify expressions. One important identity is the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).
These identities help in transforming complicated expressions into manageable forms. Take 8a^3 + 1, for example. We identified it as (2a)^3 + (1)^3. By recognizing this, we quickly applied the sum of cubes identity to simplify: (2a + 1)(4a^2 - 2a + 1).
Mastering algebraic identities is essential for success in algebra. They offer shortcuts and deeper understanding of problems, making solutions more accessible.
Intermediate Algebra
Intermediate algebra builds on basic algebra and prepares you for advanced math topics. It involves more complex factoring, working with polynomials, and applying identities. Context, such as recognizing forms like sums of cubes, becomes integral. For the expression 8a^3 + 1, intermediate algebra techniques helped us break it down and factor it: (2a)^3 + (1)^3.
By intermediate level, aim to be comfortable using formulas, identities, and systematic approaches to solving equations. In simpler terms, understand the 'why' behind steps. This example with sum of cubes showcases how deeper understanding aids problem-solving. Practicing and reinforcing these concepts prepares you for higher-level math challenges.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
Most popular questions from this chapter
Find the domain of the function \(f\) given by each of the following. $$f(x)=\frac{2}{x^{2}-7 x+10}$$Find a linear function whose graph has slope \(-\frac{1}{2}\) and contains\((3,7) .\)Mary Louise is attempting to solve \(x^{3}+20 x^{2}+4 x+80=0\) with a graphingcalculator. Unfortunately, when she graphs \(y_{1}=x^{3}+20 x^{2}+4 x+80\) in astandard \([-10,10,-10,10]\) window, she sees no graph at all, let alone any \(x\)-intercept. Can this problem be solved graphically? If so, how? If not, whynot?Find the domain of the function \(f\) given by each of the following. $$f(x)=\frac{1+x}{3 x-15 x^{2}}$$Solve. The foot of an extension ladder is 10 ft from a wall. The ladder is 2 ftlonger than the height that it reaches on the wall. How far up the wall doesthe ladder reach?
See all solutions
Recommended explanations on Math Textbooks
Mechanics Maths
Read Explanation
Applied Mathematics
Read Explanation
Logic and Functions
Read Explanation
Probability and Statistics
Read Explanation
Calculus
Read Explanation
Theoretical and Mathematical Physics
Read Explanation
View all explanations
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
In LateX if we want to insert a widehat over some word we just need to write: \widehat{ABCD}. In Mathematica, as far as I can see, we can only insert: OverHat[ABCD] , which does not puts really a widehat, but a small hat in the middle of the word..
The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It's easy to explain: take any positive number, if it's even, divide it by 2; if it's odd, multiply it by 3 and add 1. Repeat this process with the resulting number, and the conjecture says that you will eventually reach 1.
In order to put a hat on a character that is to be displayed within a figure, the "Interpreter" property of the text should to be set to "LaTeX" and the equivalent LaTeX command "\hat{x}" should be typed as "$$\hat{x}$$". where "xp" and "yp" determine the position of the displayed text.
Formula: Mass/volume (%)= mass of solute (g) volume of solution (mL)×100. Calculations: Mass volume (%)=50 g glucose 1000 mL solution ×100=5.0% glucose solution by mv. The two conversion factors from ms/v % concentration are: given g solute 100 mL solution and 100 mL solution given g solute.
Bring the variable terms to one side of the equation and the constant terms to the other side using the addition and subtraction properties of equality. Make the coefficient of the variable as 1, using the multiplication or division properties of equality. isolate the variable and get the solution.
Inline mode - Put $$ around your LaTeX markup, and Gradescope will render your math expressions inline with text.Display (Paragraph) mode - Put $$$ delimiters around your LaTeX markup. Your math expressions will appear larger and in a separate paragraph.
The command \includegraphics[scale=1.5]{overleaf-logo} will include the image overleaf-logo in the document, the extra parameter scale=1.5 will do exactly that, scale the image 1.5 of its real size. You can also scale the image to a some specific width and height.
1. Riemann Hypothesis. The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is a central problem in number theory, and discusses the distribution of prime numbers. The hypothesis focuses on the zeros of the Riemann zeta function.
Multiply by 3 and add 1. From the resulting even number, divide away the highest power of 2 to get a new odd number T(x). If you keep repeating this operation do you eventually hit 1, no matter what odd number you began with? Simple to state, this problem remains unsolved.
∼ - this symbol means “similar to”. ≈ - this symbol means “approximately”, which means something is almost, but not quite, equal to something else. ∥ - parallel.
In mathematics, the tilde operator (which can be represented by a tilde or the dedicated character U+223C ∼ TILDE OPERATOR), sometimes called "twiddle", is often used to denote an equivalence relation between two objects. Thus "x ~ y" means "x is equivalent to y". It is a weaker statement than stating that x equals y.
Bar or Vinculum: When the line above the letter represents a bar. A vinculum is a horizontal line used in the mathematical notation for a specific purpose to indicate that the letter or expression is grouped together.
If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions..
A solution is any value of a variable that makes the specified equation true. A solution set is the set of all variables that makes the equation true. The solution set of 2y + 6 = 14 is {4}, because 2(4) + 6 = 14. The solution set of y [ 2 ] + 6 = 5y is {2, 3} because 2 [ 2 ] + 6 = 5(2) and 3 [ 2 ] + 6 = 5(3).
Introduction: My name is Rueben Jacobs, I am a cooperative, beautiful, kind, comfortable, glamorous, open, magnificent person who loves writing and wants to share my knowledge and understanding with you.
We notice you're using an ad blocker
Without advertising income, we can't keep making this site awesome for you.
![Problem 18 Factor completely. $$ 8 a^{3... [FREE SOLUTION] (3)](https://i0.wp.com/website-cdn.studysmarter.de/sites/14/2023/12/Rectangle-22721-2.png "Problem 18 Factor completely. $$ 8 a^{3... [FREE SOLUTION] (3)")
![Problem 18 Factor completely.
$$
8 a^{3... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)