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Chapter 5: Problem 110

Factor by grouping. \(b^{3}-2+5 a b^{3}-10 a\)

Short Answer

Expert verified
(b^{3} -2)(1 +5a)

Step by step solution

01

- Group the terms

Group the terms in pairs that have common factors. For the given expression, group the first two terms and the last two terms: equation: (b^{3} + 5a b^{3}) + (-2 - 10a)
02

- Factor out the greatest common factor (GCF) from each group

Factor out the GCF from each pair. equation: b^{3}(1 + 5a) - 2(1 + 5a)
03

- Factor out the common binomial factor

Factor out the common binomial factor (1 + 5a) from each group: equation: (b^{3} - 2)(1 + 5a)

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Key Concepts

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

Greatest Common Factor (GCF)
When working with algebraic expressions, the Greatest Common Factor (GCF) plays a key role in simplifying and factoring expressions.
The GCF is the highest number or expression that evenly divides each term in a polynomial.
To find the GCF of multiple terms, follow these steps:
  • List the factors of each term.
  • Identify the common factors.
  • Select the highest common factor shared by all terms.
In the given expression, we have two groups: \( b^{3} + 5ab^{3} \) and \( -2 - 10a \).
For \( b^{3} + 5ab^{3} \), the GCF is \( b^{3} \).
For \( -2 - 10a \), the GCF is \( -2 \).
So, we factor out \( b^{3} \) from the first group and \( -2 \) from the second group.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations such as addition and multiplication.
Expressions can be simple like \( 3x + 4 \) or complex like \( b^{3}-2+5ab^{3}-10a \).
To simplify and solve algebraic expressions:
  • Combine like terms.
  • Use the distributive property.
  • Apply factoring techniques like grouping.
Grouping helps to manage complex expressions by creating smaller, more manageable segments.
In our example, we grouped the terms as \( (b^{3} + 5ab^{3}) + (-2 - 10a) \) to make factoring easier.
This is a crucial step to set up the factoring by grouping technique.
Polynomial Factoring
Factoring polynomials involves writing a polynomial as a product of its factors.
One effective method is 'factoring by grouping'.
Here are the steps for this method, illustrated with our example:
  • Group terms with common factors: \( (b^{3} + 5ab^{3}) + (-2 - 10a) \).
  • Factor out the GCF from each group: \( b^{3}(1 + 5a) - 2(1 + 5a) \).
  • Identify the common binomial factor in the groups: \( (1 + 5a) \).
  • Factor out the common binomial factor: \( (b^{3} - 2)(1 + 5a) \).
Factoring helps simplify expressions, making them easier to work with, whether solving equations or simplifying further.
This technique is useful in higher levels of algebra, calculus, and beyond.

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Most popular questions from this chapter

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