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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1:32 minutes

Problem 35

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 4a² + 12x + 36

Verified step by step guidance

Step 1: Begin by rearranging the terms of the polynomial to group them in a way that might reveal a common factor or pattern. Consider the expression: \(x^2 + 12x - 4a^2 + 36\).

Step 2: Notice that the first three terms \(x^2 + 12x + 36\) form a perfect square trinomial. Rewrite it as \((x + 6)^2\).

Step 3: Substitute the perfect square trinomial back into the expression, resulting in \((x + 6)^2 - 4a^2\).

Step 4: Recognize that the expression is now in the form of a difference of squares, \((A^2 - B^2)\), where \(A = (x + 6)\) and \(B = 2a\).

Step 5: Apply the difference of squares formula \(A^2 - B^2 = (A - B)(A + B)\) to factor the expression as \((x + 6 - 2a)(x + 6 + 2a)\).

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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 expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor, using special products like the difference of squares, and applying the quadratic formula when necessary.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically in the form ax² + bx + c. Understanding the structure of quadratic expressions is crucial for factoring, as it allows one to identify potential roots and apply methods such as completing the square or using the quadratic formula. Recognizing patterns in quadratics can also facilitate easier factoring.

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials with real coefficients. Identifying whether a polynomial is prime is important in algebra, as it determines the methods available for solving equations. A polynomial may be prime if it does not have rational roots or if it cannot be expressed in simpler polynomial forms.