Textbook Question
In Exercises 23–48, factor completely, or state that the polynomial is prime.x⁴ - 16
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0. Review of Algebra
Factoring Polynomials
2:18 minutesProblem 41
Textbook Question
In Exercises 35–44, factor the greatest common binomial factor from each polynomial.4x²(3x−1) + 3x − 1
Verified step by step guidance1
Identify the common binomial factor in the expression: \(4x^2(3x-1) + 3x - 1\).
Notice that the binomial \((3x - 1)\) appears in both terms: \(4x^2(3x-1)\) and \(3x - 1\).
Factor out the common binomial \((3x - 1)\) from the entire expression.
Rewrite the expression as \((3x - 1)(4x^2 + 1)\).
Verify the factorization by expanding \((3x - 1)(4x^2 + 1)\) to ensure it equals the original expression.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original polynomial. This process is essential for simplifying expressions and solving equations. In this case, recognizing common factors within the polynomial helps in rewriting it in a more manageable form.
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Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or expressions without leaving a remainder. In polynomial expressions, identifying the GCF allows for the extraction of common terms, simplifying the polynomial and making it easier to work with. This is crucial for factoring out the common binomial factor in the given expression.
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Binomial Expressions
A binomial expression is a polynomial that consists of exactly two terms, which can be separated by addition or subtraction. Understanding binomials is important for factoring, as they often represent the simplest form of polynomials. In the context of the given problem, recognizing the binomial factor is key to simplifying the polynomial expression effectively.
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