Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

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1:47 minutes

Problem 47

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x³y − 16xy³

Verified step by step guidance

Identify the greatest common factor (GCF) of the terms in the polynomial. In this case, the GCF is \( xy \).

Factor out the GCF \( xy \) from each term in the polynomial. This gives you \( xy(x^2 - 16y^2) \).

Notice that the expression inside the parentheses, \( x^2 - 16y^2 \), is a difference of squares.

Recall the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \).

Apply the difference of squares formula to \( x^2 - 16y^2 \), where \( a = x \) and \( b = 4y \), to factor it further.

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Key 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 expression into simpler components, or factors, that when multiplied together yield the original polynomial. This process often requires identifying common factors, applying special factoring techniques such as difference of squares, or using methods like grouping. Understanding how to factor is essential for simplifying expressions and solving polynomial equations.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. To factor a polynomial, identifying the GCF is often the first step, as it can simplify the expression significantly. For the polynomial x³y − 16xy³, recognizing that both terms share a common factor of xy allows for easier factoring and simplification.

Difference of Squares

The difference of squares is a specific factoring technique used when a polynomial can be expressed in the form a² - b², which factors into (a - b)(a + b). In the given polynomial, after factoring out the GCF, the remaining expression can be analyzed for patterns that fit this form, allowing for further simplification. Recognizing this pattern is crucial for complete factorization.