Problem 252 Factor. \(v^{3}-216\)... [FREE SOLUTION]

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Chapter 7: Problem 252

Factor. \(v^{3}-216\)

Short Answer

Expert verified

\((v - 6)(v^2 + 6v + 36)\)

Step by step solution

Identify the form

The given expression is a difference of cubes, which has the form \(a^3 - b^3\).

Rewrite the expression

Rewrite the expression in terms of cubic terms: \(v^3 - 216\) can be written as \(v^3 - 6^3\).

Apply the difference of cubes formula

Use the formula for factoring the difference of cubes, which is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(a = v\) and \(b = 6\).

Substitute and simplify

Substitute \(a\) and \(b\): \((v - 6)(v^2 + 6v + 36)\).

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Key Concepts

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

difference of cubes

To factor the expression \(v^3 - 216\), we first recognize the pattern known as the difference of cubes. This is a special type of polynomial where one term is subtracted from another, and both are cubes. In general, the formula for factoring a difference of cubes is:

\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]

Here, when identifying \(a\) and \(b\), we have:

- \(a = v\) (since \(v^3 = a^3\))

- \(b = 6\) (since \(216 = 6^3\))

Substituting these values into the difference of cubes formula, the factorization becomes:

\[(v - 6)(v^2 + 6v + 36)\]

Recognizing and applying this pattern simplifies the factorization process.

algebraic expressions

Algebraic expressions involve variables, numbers, and arithmetic operations. Simplifying or factoring them is a fundamental skill in algebra. Let鈥檚 consider the expression we factored earlier, \(v^3 - 216\).

When working with algebraic expressions:

Identify patterns and forms (like the difference of cubes).
Rewrite terms to expose the structure (\(v^3 - 216\) to \(v^3 - 6^3\)).
Use known formulas and identities to simplify or factor (such as the difference of cubes formula).

Mastering these skills allows you to tackle more complex algebraic problems effectively.

polynomial identities

Polynomial identities are equations that hold true for any value of the variable within the polynomial. They are immensely useful in algebra for simplifying, factoring, and solving equations.

The difference of cubes is just one of many important polynomial identities. Here's a brief reminder of a few others:

The sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
The difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).

By understanding and memorizing these identities, you can quickly factor various polynomial expressions and solve complex algebraic problems more efficiently. For example, using the difference of cubes identity, we turned \(v^3 - 216\) into a product of simpler polynomials: \[(v - 6)(v^2 + 6v + 36)\].

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91影视

Factor. \(v^{3}-216\)

Short Answer

Step by step solution

Identify the form

Rewrite the expression

Apply the difference of cubes formula

Substitute and simplify

Key Concepts

difference of cubes

algebraic expressions

polynomial identities

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