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Chapter 7: Problem 43

Factor the greatest common factor from each polynomial. \(3 b(b-2)-13(b-2)\)

Short Answer

Expert verified
(b-2)(3b - 13)

Step by step solution

01

Identify Common Factors in Each Term

Look at each term in the polynomial. The two terms are \(3b(b-2)\) and \(-13(b-2)\). Observe that \((b-2)\) is present in both terms.
02

Factor Out (b-2)

Since \((b-2)\) is common in both terms, factor it out. Write the polynomial as \( (b-2)(3b - 13) \).

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Key Concepts

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

Factoring Polynomials
To understand factoring, let's break it down step by step.
Polynomials are mathematical expressions that consist of variables and coefficients.
Factoring a polynomial involves rewriting it as a product of simpler polynomials.
Consider the polynomial from our exercise: \(3b(b-2) - 13(b-2)\).
The goal is to express it in a factored form. Factoring helps simplify polynomials, making them easier to work with in calculations.
Common Factors
When working with polynomials, always look for common factors in each term.
In the polynomial from the exercise, the terms are \(3b(b-2)\) and \(-13(b-2)\).
Both terms clearly have a common factor: \( (b-2) \).
A common factor is a term that appears in all the terms of the polynomial. By identifying the common factor, you can factor it out and simplify the polynomial. This technique is crucial in polynomial operations and solving equations.
Polynomial Expressions
Polynomial expressions are a sum of multiple terms consisting of variables and coefficients.
They can vary in complexity from simple expressions like \(x^2 + 3x + 2\) to more complex ones like \(3b(b-2)-13(b-2)\).
Understanding how to manipulate and factor these expressions is fundamental in algebra.
By factoring polynomials, you simplify the expressions.
In our exercise, factoring out the common factor \((b-2)\), transforms the polynomial into a more manageable form: \((b-2)(3b - 13)\).
This makes it easier to solve or further simplify the expression.
Each step helps you gain better control over the polynomial, paving the way for more advanced math concepts.

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Most popular questions from this chapter

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