Question

Factor completely. $$ x^{3}-7 x^{2} y+12 x y^{2} $$

   Factor completely.
$$
x^{3}-7 x^{2} y+12 x y^{2}
$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Margaret L. Lial,… 7th Edition

Chapter 5, Problem 69

Instant Answer

Solved by Expert Gregory Higby

02/09/2022

Step 1

The common factor here is $x$. So, we factor out $x$ from each term. This gives us: $$ x(x^{2}-7 x y+12 y^{2}) $$ Show more…

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Factor completely. $$ x^{3}-7 x^{2} y+12 x y^{2} $$

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Key Concepts

Polynomial Factorization Techniques

Polynomial factorization is the process of rewriting a polynomial as a product of simpler polynomials or monomials. This process is crucial for simplifying expressions, solving equations, and understanding the structural properties of polynomials. Techniques include extracting the greatest common factor, grouping, and applying the product-sum method for quadratics.

Extracting the Greatest Common Factor

Before attempting further factorization, it is important to look for and remove the greatest common factor (GCF) that is present in all terms. This step simplifies the expression and often makes the remaining factors more apparent.

Factoring Quadratic Polynomials

Once the GCF has been factored out, the residual expression may be a quadratic polynomial. Factoring quadratics involves finding two numbers that multiply to give the product of the quadratic’s leading coefficient and constant term, while also summing to the middle coefficient. This concept is fundamental in breaking down more complex polynomial expressions.

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