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Chapter 6: Problem 12

Factor completely. $$ t^{3}+27 $$

Short Answer

Expert verified
\((t + 3)(t^2 - 3t + 9)\)

Step by step solution

01

Identify the expression type

The given expression is a sum of cubes, which can be identified as the expression in the form of \(a^3 + b^3\).
02

Rewrite the expression

Rewrite the given expression in the form of a sum of cubes: \(t^3 + 27\). Notice that \(27\) can be written as \((3)^3\), so it becomes \(t^3 + 3^3\).
03

Apply the sum of cubes formula

The sum of cubes formula is given by \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = t\) and \(b = 3\).
04

Substitute the values

Substitute \(t\) and \(3\) into the sum of cubes formula: \[(t + 3)(t^2 - t \cdot 3 + 3^2)\].
05

Simplify the expression

Finally, simplify the expression inside the parentheses: \(t^2 - 3t + 9\).This gives the factorized form of the original expression as \((t + 3)(t^2 - 3t + 9)\).

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Key Concepts

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

Sum of Cubes
A key component of this exercise is understanding the sum of cubes. The sum of cubes refers to an algebraic expression that is in the form of \(a^3 + b^3\). It's called the sum of cubes because both terms are perfect cubes. For instance, in \(t^3 + 27\), \(t^3\) is the cube of \(t\), and \(27\) is the cube of \(3\). Recognizing this pattern allows us to use a special factorization formula to break it down further.
Polynomial Factorization
Factorization involves breaking down a polynomial into simpler polynomials whose product is the original polynomial. In this exercise, we identify the given expression as the sum of cubes and then rewrite it accordingly.
Step-by-step, we applied the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = t\) and \(b = 3\). When we substitute these values, we get \(t^3 + 3^3 = (t + 3)(t^2 - 3t + 9)\). This results in the factorized form of the expression, making it simpler to work with or solve.
Remember, the goal is to express the polynomial in a product form, which often makes solving equations or simplifying expressions much easier.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. In our example, \(t^3 + 27\) is an algebraic expression where we apply factorization.
Here are some key points for working with algebraic expressions:
  • Identify the type of expression (like sum of cubes in this case).
  • Rewrite the expression in a recognizable form.
  • Use algebraic identities and formulas to factorize or simplify.
By recognizing \(t^3 + 27\) as a sum of cubes, we utilized the sum of cubes formula to simplify it to \( (t + 3)(t^2 - 3t + 9) \). Understanding these processes is crucial in algebra as it makes complex problems more manageable.

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Most popular questions from this chapter

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{2}-\left(\frac{1}{x}\right)^{2} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{2 a}-y^{2} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ a^{2}+1=2 a $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}-9=0 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 x^{4}-25 x^{2} $$
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