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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1:59 minutes

Problem 59

Textbook Question

Factor each polynomial. See Examples 5 and 6. 9a^2-16

Verified step by step guidance

Identify the expression as a difference of squares: $9a^2 - 16$.

Recall the formula for factoring a difference of squares: $x^2 - y^2 = (x + y)(x - y)$.

Recognize that $9a^2$ is a perfect square, specifically $(3a)^2$, and $16$ is a perfect square, specifically $4^2$.

Apply the difference of squares formula: $(3a)^2 - 4^2 = (3a + 4)(3a - 4)$.

Write the factored form of the polynomial: $(3a + 4)(3a - 4)$.

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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 rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and understanding the polynomial's roots. Common methods include factoring out the greatest common factor, using special products, and applying techniques like grouping.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions in the form a^2 - b^2, which can be factored as (a - b)(a + b). In the given polynomial 9a^2 - 16, both 9a^2 and 16 are perfect squares, making this pattern applicable. Recognizing this pattern is crucial for efficient factoring.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer or a variable. In the context of polynomials, terms like 9a^2 (which is (3a)^2) and 16 (which is 4^2) are perfect squares. Identifying perfect squares is vital for applying the difference of squares method effectively in polynomial factoring.