Factor each trinomial, if possible. See Examples 3 and 4. 36x³+18x²-4x

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First, identify the greatest common factor (GCF) of all the terms in the trinomial \(36x^3 + 18x^2 - 4x\). Look at the coefficients (36, 18, and 4) and the variable parts (\(x^3\), \(x^2\), and \(x\)) to find the GCF.

Factor out the GCF from each term. This means you will write the expression as the product of the GCF and a new trinomial inside parentheses.

After factoring out the GCF, focus on the trinomial inside the parentheses. Check if this trinomial can be factored further by looking for two numbers that multiply to the product of the first and last coefficients and add to the middle coefficient.

If the trinomial inside the parentheses is factorable, rewrite it as the product of two binomials. If it is not factorable, leave it as is.

Write the final factored form as the product of the GCF and the factored trinomial (or the original trinomial if it cannot be factored further).

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Key Concepts

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

Factoring Out the Greatest Common Factor (GCF)

Factoring begins by identifying the greatest common factor shared by all terms in the polynomial. Extracting the GCF simplifies the expression and makes further factoring easier. For example, in 36x^3 + 18x^2 - 4x, the GCF is 2x.

Factoring Trinomials

A trinomial is a polynomial with three terms, often factored into the product of binomials. After factoring out the GCF, the remaining trinomial can be factored by finding two numbers that multiply to the product of the leading coefficient and constant term and add to the middle term's coefficient.

Polynomial Degree and Terms

Understanding the degree of each term and the overall polynomial helps determine the factoring strategy. Here, the polynomial is cubic (degree 3), but factoring out the GCF reduces it to a quadratic trinomial, which is easier to factor using standard methods.

Factor each trinomial, if possible. See Examples 3 and 4. 36x3+18x2-4x