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Chapter 5: Problem 185

Factor each polynomial completely. $$5 b^{3}-40$$

Short Answer

Expert verified
5(b - 2)(b^2 + 2b + 4)

Step by step solution

01

Identify Common Factors

The first step is to look for common factors in each term of the polynomial. Here, both terms, 5 and 40, are divisible by 5. So, factor out the greatest common factor (GCF), which is 5. This gives us: 5(b^3 - 8).
02

Recognize the Difference of Cubes

Observe that the expression inside the parentheses, b^3 - 8 , can be written as a difference of cubes. Rewrite 8 as 2^3 to get: b^3 - 2^3.
03

Apply the Difference of Cubes Formula

Use the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2) Here, a is b and b is 2. Applying this formula, we get: (b - 2)(b^2 + 2b + 4).
04

Write the Final Factored Form

Combine the GCF from Step 1 and the factored form from Step 3 to get the final factored expression: 5(b - 2)(b^2 + 2b + 4).

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Key Concepts

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

Greatest Common Factor
When beginning to factor any polynomial, the first step is to look for the Greatest Common Factor (GCF). This is the largest factor that divides all terms in the polynomial without leaving a remainder. By factoring out the GCF, you simplify the polynomial, making further factorization easier.

An example:
To factor the polynomial 5b鲁 - 40, we notice that both terms (5b鲁 and 40) are divisible by 5. Therefore, the GCF is 5. We factor out the 5 to get:
\[5(b^3 - 8)\]
Understanding GCF is critical as it helps reduce the polynomial to its simplest form before further factorization.
Difference of Cubes
A 'difference of cubes' occurs when you have an expression of the form \(a^3 - b^3\). This type of expression can be factored using a specific formula:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
Recognizing this pattern in polynomials is key. For example, in the expression \(b^3 - 8\), you can rewrite 8 as \(2^3\), making it fit the difference of cubes formula. Here, \(a\) is \(b\) and \(b\) is 2. Applying the formula, we get:
\[b^3 - 2^3 = (b - 2)(b^2 + 2b + 4)\]
This transformation simplifies the polynomial, making it easier to work with and solve.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its factors. These factors could be numbers, variables, or polynomials themselves. Here are key steps to follow:
  • Identify the GCF and factor it out as the first step.
  • Look for special factoring patterns such as the difference of squares, difference of cubes, and perfect square trinomials.
  • If no specific pattern is apparent, try other factoring techniques such as grouping.

In our example, after factoring out the GCF and applying the difference of cubes formula, combine the results to get the final factored form:
\[5(b - 2)(b^2 + 2b + 4)\].
By mastering these steps, you can tackle the most complex polynomials with ease.

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