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Chapter 0: Problem 57

Factor using the formula for the sum or difference of two cubes. $$x^{3}+27$$

Short Answer

Expert verified
The factorized form of \(x^{3}+27\) is \((x+3)(x^{2}-3x+9)\)

Step by step solution

01

Identify a and b

In the given expression \(x^{3}+27\), 'a' can be identified as 'x' because \(a^3=x^{3}\) and 'b' can be identified as '3' because \(b^3=27\). Hence, a=x and b=3.
02

Apply the formula

Substituting 'a' and 'b' into the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), we get, \(x^{3}+27 = (x+3)(x^{2}-3x+9)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Cubes
When dealing with the sum of cubes, you aim to factor expressions of the form \(a^3 + b^3\). This formula is essential in algebra, making it easier to simplify complex expressions. In our example, the expression \(x^3 + 27\) is a sum of cubes, where we identify \(a = x\) and \(b = 3\).
To factor such expressions, we use the identity:
  • \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Plugging in our values \(a = x\) and \(b = 3\), the sum of cubes formula transforms the expression into \((x + 3)(x^2 - 3x + 9)\). Each term in the formula serves a purpose:
  • \(a + b\) helps consolidate the base terms \(a\) and \(b\).
  • \(a^2 - ab + b^2\) expands the relationship by covering all potential interactions between \(a\) and \(b\).
This breakdown not only simplifies your solution but also enhances your understanding of how elements interact in polynomial expressions.
Difference of Cubes
The difference of cubes is another polynomial identity that allows you to factor expressions like \(a^3 - b^3\). While our original problem did not involve a difference of cubes, understanding this concept is equally important.
The expression \(a^3 - b^3\) is factored using:
  • \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
What you notice here is the switch in signs. The initial parenthesis changes from addition to subtraction, and each component within it reflects this change by modifying the relationships between the terms:
  • \(a - b\) pools the base terms into a difference.
  • \(a^2 + ab + b^2\) accounts for all interactions, yet keeps each term positive, which makes the structure neat and predictable.
Keeping both sum and difference of cubes equations handy prepares you to tackle a variety of polynomial expressions with confidence.
Polynomial Identities
Polynomial identities like the sum and difference of cubes are crucial tools in algebra. They provide students with shortcuts for factoring and simplifying complex equations, saving time and avoiding the more tedious methods of expansion.
Why are polynomial identities so beneficial?
  • Efficiency: They allow you to rewrite expressions in simpler forms quickly.
  • Pattern Recognition: Mastery over these identities helps in recognizing patterns which is crucial for solving higher-level algebra problems.
  • Error Reduction: Following these established formulas minimizes the likelihood of mistakes in arithmetic or algebraic manipulation.
With polynomial identities, you handle expressions systematically, ensuring a deeper comprehension of algebra and boosting your overall problem-solving skills. Therefore, investing time in memorizing and applying these formulas will greatly enhance your mathematical journey.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned}2 &>1 \\\2(y-x) &>1(y-x) \\\2 y-2 x &>y-x \\\y-2 x &>-x \\\y &>x\end{aligned}$$ This is a true statement. Multiply both sides by \(y-x\) Use the distributive property. Subtract \(y\) from both sides. Add \(2 x\) to both sides. The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school. (GRAPH CAN NOT COPY) The data displayed by the bar graph can be described by the mathematical model $$p=\frac{4 x}{5}+25$$ where \(x\) is the number of years after 1980 and \(p\) is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information a. According to the formula, in 2010 , what percentage of U.S. college freshmen had an average grade of \(A\) in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much? b. If trends shown by the formula continue, project when \(57 \%\) of U.S. college freshmen will have had an average grade of A in high school.

In Exercises \(112-123,\) use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
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