Factor each trinomial, if possible. See Examples 3 and 4. $24 a^{4} + 10 a^{3} b - 4 a^{2} b^{2}$

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First, identify the greatest common factor (GCF) of all the terms in the trinomial \$24a^4 + 10a^3b - 4a^2b^2$. Look at the coefficients (24, 10, and -4) and the variable parts ($a^4$, $a^3b$, $a^2b^2$) to find the GCF.

Factor out the GCF from each term. This means dividing each term by the GCF and writing the expression as the GCF multiplied by the resulting trinomial.

After factoring out the GCF, focus on factoring the remaining trinomial inside the parentheses. The trinomial will be of the form $Ax^2 + Bxy + Cy^2$, where you can use methods such as factoring by grouping or the AC method.

To factor the trinomial inside the parentheses, find two numbers that multiply to $A \times C$ and add to $B$. Use these numbers to split the middle term and then factor by grouping.

Write the fully factored form as the product of the GCF and the two binomials obtained from factoring the trinomial inside the parentheses.

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Key Concepts

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

Factoring Trinomials

Factoring trinomials involves expressing a three-term polynomial as a product of two binomials or other simpler polynomials. The goal is to rewrite the expression in a form that, when multiplied, returns the original trinomial. Recognizing patterns and using methods like trial and error or grouping helps in this process.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Factoring out the GCF simplifies the polynomial and makes further factoring easier. For example, in 24a^4 + 10a^3b - 4a^2b^2, the GCF includes powers of 'a' and numerical factors.

Factoring by Grouping

Factoring by grouping involves splitting a polynomial into groups of terms that share common factors, then factoring each group separately. This method is useful when a trinomial or polynomial does not factor easily by standard methods. After grouping, common binomial factors can be factored out to simplify the expression.

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