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Chapter 0: Problem 90

Factor completely, or state that the polynomial is prime. $$ 12 x^{2} y-27 y-4 x^{2}+9 $$

Short Answer

Expert verified
The completely factored form of the given polynomial is: \((3y - 1)(2x + 3)(2x - 3)\)

Step by step solution

01

Rearrange the terms

First, rearrange the terms of the given polynomial such that like terms are beside each other. Rearrange the terms so that the terms with the same variable are together. This will look like:\[ 12 x^{2} y - 4 x^{2} + (-27y + 9) \]
02

Factor out common terms

Next, factor out the terms that are common to the two groups we formed. For the first two terms, we can factor out \(4 x^{2}\) to get \(4 x^{2}(3y - 1)\), and from the last two terms, we can factor out 9, to get \(9(-3y+1)\). So the polynomial becomes:\[ 4 x^{2}(3y - 1) - 9(3y - 1) \]
03

Factor out the common binomial

Now, check if there is any binomial common. Here, \(3y - 1\) is a common binomial. We can factor that out to get:\[ (3y - 1)(4x^{2} - 9) \]
04

Further factorization

The right parenthesis in the factored result looks like a difference of squares. A difference of squares could be factored into \((a+b)(a-b)\). Therefore, we could write \(4x^{2} - 9) as \((2x)^{2} - 3^{2}\), which could then be factored to get: \((2x+3)(2x-3)\)\[ (3y - 1)(2x + 3)(2x - 3) \]

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Key Concepts

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

Difference of Squares
Understanding the concept of the difference of squares is crucial when factoring certain types of polynomial expressions. It refers to an expression of the form \( a^2 - b^2 \), which can be factored into \( (a + b)(a - b) \). This factorization is possible because when \( a + b \) and \( a - b \) are multiplied, the middle terms cancel out, leaving us with \( a^2 - b^2 \).

For instance, if we consider \( 4x^2 - 9 \) from the given exercise, we can recognize it as a difference of squares because it takes the form of \( a^2 - b^2 \) where \( a = 2x \) and \( b = 3 \). Factoring this gives us \( (2x + 3)(2x - 3) \), demonstrating the use of this concept in breaking down more complex algebraic expressions.
Common Binomial Factor
When factoring polynomials, it's often helpful to look for a common binomial factor among the terms. A binomial is an algebraic expression that contains exactly two terms. If the same binomial appears in each term of a polynomial, we can factor it out, simplifying the expression.

In the given exercise, after rearranging the terms, we can spot the common binomial factor \( 3y - 1 \). This binomial appears both in \( 4x^2(3y - 1) \) and \( -9(3y - 1) \), indicating that we can factor \( 3y - 1 \) out of the expression. The result is \( (3y - 1)(4x^2 - 9) \), clearly showing the importance of identifying common binomial factors to simplify polynomial expressions.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations. These expressions can be simplified or manipulated using various algebraic principles, including factoring. Factoring involves breaking down a complex expression into simpler, multipliable components, or 'factors,' that, when multiplied, give back the original expression.

In our exercise, we begin with the algebraic expression \( 12x^2y - 27y - 4x^2 + 9 \). By rearranging and grouping the terms, looking for common factors, and utilizing our knowledge of the difference of squares, we ultimately factor the expression completely. The process involves understanding how these expressions are structured and recognizing patterns that allow us to simplify them, an essential skill in the study of algebra.

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Most popular questions from this chapter

Use Einstein's special-relativity equation $$R_{a}=R_{f} \sqrt{1-\left(\frac{v}{c}\right)^{2}}$$ described in the Blitzer Bonus on page \(47,\) to solve this exercise. You are moving at \(90 \%\) of the speed of light. Substitute \(0.9 c\) for \(v,\) your velocity, in the equation. What is your aging rate, correct to two decimal places, relative to a friend on Earth? If you are gone for 44 weeks, approximately how many weeks have passed for your friend?

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

Evaluate each expression. $$\sqrt[3]{\sqrt[4]{16}+\sqrt{625}}$$

Describe the kinds of numbers that have rational fifth roots.
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