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Chapter 5: Problem 51

Factor each trinomial. \(2 x^{3} y^{3}-48 x^{2} y^{4}+288 x y^{5}\)

Short Answer

Expert verified
2xy^3 (x - 12y)^2

Step by step solution

01

- Identify common factors

Each term in the trinomial has a common factor. Identify the greatest common factor (GCF) among the terms. The GCF of the terms is 2, since the smallest coefficient is 2, and they all share the variables x and y. The smallest power of x is 1 and the smallest power of y is 3. Therefore, the GCF is 2xy^3.
02

- Factor out the GCF

Divide each term by the GCF and factor out the GCF: a) First term: divide 2x^3y^3 by 2xy^3: (2x^3y^3)/(2xy^3) = x^2b) Second term: divide -48x^2y^4by 2xy^3:(-48x^2y^4)/(2xy^3) = -24xyc) Third term: divide 288xy^5 by 2xy^3:(288xy^5)/(2xy^3) = 144y^2Therefore, factor out the GCF 2xy^3 from the trinomial: 2xy^3(x^2 - 24xy + 144y^2)
03

- Factor the trinomial inside the parentheses

Recognize that x^2 - 24xy + 144y^2 is a perfect square trinomial. It can be factored as: (x - 12y)^2.Therefore, the complete factorization is: 2xy^3 (x - 12y)^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
When factoring trinomials, the first step is usually to find the greatest common factor (GCF). The GCF is the largest factor that can evenly divide each term in the polynomial.

Let's consider the trinomial: \(2x^3y^3 - 48x^2y^4 + 288xy^5 \).
The GCF is found by identifying the smallest coefficients and the smallest powers of shared variables.

In this case, the smallest coefficient is 2, and the terms share the variables x and y. The smallest power of x is 1, and the smallest power of y is 3.

Therefore, the GCF is \(2xy^3 \).
factoring out the GCF
Once you identify the GCF, the next step is to factor it out of the trinomial. This means you divide each term of the polynomial by the GCF and rewrite the trinomial accordingly.

Let's divide the terms of \(2x^3y^3 - 48x^2y^4 + 288xy^5 \) by \(2xy^3 \).
  • First term: \( \frac{2x^3y^3}{2xy^3} = x^2 \)
  • Second term: \( \frac{-48x^2y^4}{2xy^3} = -24xy \)
  • Third term: \( \frac{288xy^5}{2xy^3} = 144y^2 \)
This allows us to rewrite the trinomial as: \( 2xy^3(x^2 - 24xy + 144y^2) \).
perfect square trinomial
The last step is recognizing and factoring the trinomial inside the parentheses.

Observe the expression \( x^2 - 24xy + 144y^2 \).
This type of trinomial is a perfect square trinomial. A perfect square trinomial is one that can be written in the form \( (a - b)^2 \) or \( (a + b)^2 \).

Recognizing the pattern, we see that \( x^2 - 24xy + 144y^2 \) can be factored as \( (x - 12y)^2 \).
Therefore, the complete factorization of the original trinomial is: \( 2xy^3 (x - 12y)^2 \).

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