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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 32
Chapter 1, Problem 32

Factor each polynomial by grouping. See Example 2. 4x6+36−x6y−9y4x^6+36-x^6y-9y

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Rewrite the polynomial to group terms that can be factored together: \(4x^{6} + 36 - x^{6}y - 9y\) can be grouped as \((4x^{6} + 36) - (x^{6}y + 9y)\).
Factor out the greatest common factor (GCF) from each group: from the first group \(4x^{6} + 36\), factor out \(4\) to get \(4(x^{6} + 9)\); from the second group \(x^{6}y + 9y\), factor out \(y\) to get \(y(x^{6} + 9)\).
Rewrite the expression using the factored groups: \(4(x^{6} + 9) - y(x^{6} + 9)\).
Notice that \((x^{6} + 9)\) is a common binomial factor; factor it out: \((x^{6} + 9)(4 - y)\).
The polynomial is now factored by grouping as \((x^{6} + 9)(4 - y)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Grouping

Polynomial grouping involves rearranging and grouping terms in a polynomial to factor each group separately. This method is useful when a polynomial has four or more terms and can be split into pairs or groups that share common factors.
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Factoring Common Factors

Factoring common factors means identifying and extracting the greatest common factor (GCF) from each group of terms. This simplifies the expression and helps reveal a common binomial factor, which is essential for completing the factorization by grouping.
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Difference of Squares

The difference of squares is a special factoring pattern where an expression of the form a^2 - b^2 factors into (a - b)(a + b). Recognizing this pattern helps factor terms like x^6 - 9y, where powers and constants can be expressed as squares.
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