Textbook Question
Factor each polynomial by grouping. See Example 2. 10ab-6b+35a-21
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0. Review of Algebra
Factoring Polynomials
3:36 minutesProblem 32
Textbook Question
Factor each polynomial by grouping. See Example 2. 4x^6+36-x^6y-9y
Verified step by step guidance1
Group the terms in pairs: \((4x^6 + 36) + (-x^6y - 9y)\).
Factor out the greatest common factor from each pair: \(4(x^6 + 9) - y(x^6 + 9)\).
Notice that \((x^6 + 9)\) is a common factor in both terms.
Factor out the common factor \((x^6 + 9)\): \((x^6 + 9)(4 - y)\).
Verify the factorization by expanding \((x^6 + 9)(4 - y)\) to ensure it equals the original polynomial.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials by rearranging and grouping terms in pairs or sets. This technique involves identifying common factors within these groups, allowing for simplification of the polynomial. It is particularly useful when dealing with polynomials that have four or more terms, as it can reveal factors that are not immediately obvious.
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Common Factors
A common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of polynomials, identifying common factors is crucial for simplifying expressions and factoring. By extracting the greatest common factor (GCF) from grouped terms, one can simplify the polynomial and make it easier to factor further or solve.
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Polynomial Structure
Understanding the structure of a polynomial is essential for effective factoring. A polynomial is an expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Recognizing the degree of each term and the overall polynomial helps in determining the best approach for factoring, including grouping, and can guide the identification of potential factors.
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