Problem 109 Factor by grouping. \(a^{5}-3+... [FREE SOLUTION]

Chapter 5: Problem 109

Factor by grouping. \(a^{5}-3+2 a^{5} b-6 b\)

Short Answer

Expert verified

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

Step by step solution

- Group Terms

Group the terms in pairs to make it easier to factor by grouping: a^{5} - 3 + 2a^{5}b - 6beq (a^{5} + 2a^{5}b) - (3 + 6b)

- Factor out Common Factors in Each Group

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

- Factor out the Common Binomial Factor

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

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Key Concepts

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

polynomial factoring

Polynomial factoring is an essential skill in algebra that involves breaking down a polynomial into simpler 鈥榝actor鈥� polynomials that can be multiplied to get the original polynomial. This process can make solving equations easier and can help to understand the properties of a polynomial.
For example, the polynomial in the exercise, a^{5}-3+2 a^{5} b-6 b, can be factored using a technique called 鈥榝actor by grouping鈥�.
This involves breaking the polynomial into smaller groups of terms. This makes it easier to spot and take out common factors.

common factors

A common factor is a factor that is shared by all terms in a polynomial. Identifying and factoring out common factors is a key step in simplifying a polynomial.
In the exercise, we factor in steps by grouping:

First, we group terms in pairs: (a^{5} + 2a^{5}b) - (3 + 6b).
Next, we identify and factor out the common factor in each group: a^{5}(1 + 2b) - 3(1 + 2b).

Notice how each group now has a common binomial factor (1 + 2b).

binomial factors

A binomial factor is simply a factor that consists of two terms. Factoring binomials often helps to simplify and solve polynomial equations.
In our exercise, after extracting common factors and organizing the terms into groups, we can see our common binomial factor: (1 + 2b).
By pulling out this binomial factor from each group, we achieve the final factored form:

Step 3 - Factor out the Common Binomial Factor: a^{5}(1 + 2b) - 3(1 + 2b) => (1 + 2b)(a^{5} - 3).

This last step demonstrates the importance of 鈥榖inomial factors鈥� in simplifying complex polynomials.

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91影视

Factor by grouping. \(a^{5}-3+2 a^{5} b-6 b\)

Short Answer

Step by step solution

- Group Terms

- Factor out Common Factors in Each Group

- Factor out the Common Binomial Factor

Key Concepts

polynomial factoring

common factors

binomial factors

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