In Exercises 1–10, factor out the greatest common factor. 3x²+6x

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the expression. The terms are 3x^2 and 6x. The coefficients 3 and 6 have a GCF of 3, and the variable x is common to both terms with the smallest power being x^1.
2. Factor out the GCF (3x) from each term. Divide each term by the GCF to determine the remaining factors. For 3x^2, dividing by 3x leaves x, and for 6x, dividing by 3x leaves 2.
3. Write the factored form of the expression as the product of the GCF and the remaining factors. This gives 3x multiplied by the binomial (x + 2).
4. Verify your factorization by distributing the GCF back into the binomial. Multiply 3x by each term in (x + 2) to ensure it equals the original expression, 3x^2 + 6x.
5. Conclude that the factored form of the expression is correct and simplified as 3x(x + 2).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest integer or algebraic expression that divides two or more terms without leaving a remainder. To find the GCF, identify the common factors of the coefficients and the variables in each term. For example, in the expression 3x^2 and 6x, the GCF is 3x, as it is the highest factor that can be factored out from both terms.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. This is a fundamental skill in algebra, allowing for simplification and solving of equations. In the case of 3x^2 + 6x, factoring involves expressing the polynomial as a product of its GCF and another polynomial.

Polynomial Expressions

A polynomial expression is a mathematical expression that consists of variables raised to non-negative integer powers and coefficients. Polynomials can be classified by their degree and the number of terms. Understanding polynomials is crucial for operations such as addition, subtraction, multiplication, and factoring, as seen in the expression 3x^2 + 6x, which is a quadratic polynomial.