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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2:02 minutes

Problem 27

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor.−2x² + 6x − 14

Verified step by step guidance

Identify the greatest common factor (GCF) of the terms in the expression \(-2x^2 + 6x - 14\).

The GCF of the coefficients \(-2\), \(6\), and \(-14\) is \(2\).

Factor out \(-2\) from each term in the expression.

Rewrite the expression as \(-2(x^2 - 3x + 7)\).

Verify by distributing \(-2\) back through the expression to ensure it matches the original expression.

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Key Concepts

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

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest integer or algebraic expression that divides each term of a polynomial without leaving a remainder. To find the GCF, identify the common factors of the coefficients and the variables in each term. In the expression −2x² + 6x − 14, the GCF is -2, as it is the highest factor that can be factored out from all terms.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process simplifies expressions and can help solve equations. In this case, factoring out the GCF allows us to express the polynomial in a simpler form, making it easier to analyze or solve. The goal is to express the polynomial in a way that highlights its structure.

Negative Sign in Factoring

Factoring out a negative sign means that we are extracting a negative value from the polynomial, which can change the signs of the terms. This is particularly useful when the leading coefficient is negative, as it can simplify the expression and make it easier to work with. In the given polynomial, factoring out -2 not only simplifies the expression but also ensures that the leading coefficient of the resulting polynomial is positive.