Textbook Question
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² − x + 7
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0. Review of Algebra
Factoring Polynomials
2:39 minutesProblem 22
Textbook Question
Factor out the greatest common factor from each polynomial. See Example 1. 6x(a+b)-4y(a+b)
Verified step by step guidance1
Identify the greatest common factor (GCF) in the terms 6x(a+b) and -4y(a+b). Notice that both terms contain the binomial factor (a+b).
Also, look at the numerical coefficients 6 and -4. The GCF of 6 and 4 is 2, and since one term is negative, factor out +2 to keep the expression consistent.
Write the expression as a product of the GCF and a remaining binomial. The GCF includes 2 and the common binomial (a+b), so factor out 2(a+b).
Divide each term by the GCF 2(a+b) to find the remaining terms inside the parentheses: \( \frac{6x(a+b)}{2(a+b)} = 3x \) and \( \frac{-4y(a+b)}{2(a+b)} = -2y \).
Express the factored form as: \$2(a+b)(3x - 2y)$, which is the original polynomial factored by the greatest common factor.
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Video duration:
2mKey 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 expression that divides two or more terms without leaving a remainder. In polynomials, it includes the highest common numerical coefficient and any common variables with the smallest exponents. Factoring out the GCF simplifies expressions and is the first step in many factoring problems.
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Distributive Property
The distributive property states that a(b + c) = ab + ac. When factoring, this property is used in reverse to rewrite a sum of terms as a product of a common factor and a sum. Recognizing common factors in each term allows you to factor expressions efficiently.
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Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. Identifying and factoring out the GCF is often the first step, which simplifies the polynomial and makes further factoring easier. This process is essential for solving polynomial equations and simplifying expressions.
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