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Chapter 6: Problem 34

Factor completely. $$ 2 y^{3}-54 z^{3} $$

Short Answer

Expert verified
2(y - 3z)(y^2 + 3yz + 9z^2)

Step by step solution

01

Identify Common Factors

Observe the expression to see if there are any common factors in each term. In this case, both terms are divisible by 2.
02

Factor Out the Common Factor

Factor out the common factor of 2 from the expression: \[ 2 y^{3}-54 z^{3} = 2( y^{3}-27 z^{3}) \]
03

Apply the Difference of Cubes Formula

Next, recognize that \( y^3 - 27z^3 \) is a difference of cubes. Use the difference of cubes formula: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\] where \(a = y \) and \( b = 3z \).
04

Substitute and Simplify

Plug in \( y \) and \( 3z \) into the formula: \[ y^3 - (3z)^3 = (y - 3z)(y^2 + y \times 3z + (3z)^2) \] Which simplifies to: \[ y^3 - 27z^3 = (y - 3z)(y^2 + 3yz + 9z^2) \].
05

Write the Final Factored Expression

Combine the common factor from Step 2 with the factored result from Step 4: \[ 2(y^3 - 27z^3) = 2(y - 3z)(y^2 + 3yz + 9z^2) \]

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Key Concepts

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

Common Factors
When factoring polynomials, the first step is often to identify any common factors shared by the terms. A common factor is a number or variable that divides all terms in the expression without leaving a remainder.
Let's consider the original polynomial expression: \(2 y^{3}-54 z^{3}\). Here, both terms share a common numerical factor of 2. By factoring out the 2, we simplify the expression and make further steps easier: \[ 2 y^{3}-54 z^{3} = 2(y^{3}-27 z^{3}) \]. Identifying common factors helps to reduce the complexity of the given polynomial expression.
Difference of Cubes
The difference of cubes pattern is crucial for factoring certain polynomials. This pattern is expressed using the formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In this expression, the first term (\

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Most popular questions from this chapter

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{3 / 4}+x^{1 / 2}-x^{1 / 4} $$

Let \(f(x)=x^{2}-12 x .\) Find \(a\) such that \(f(a)=-36\).

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 6 x^{2}-6 y^{2} $$

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