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Chapter 5: Problem 47

Find each product. $$(x y-3)^{2}$$

Short Answer

Expert verified
The product of \( (xy - 3)^2 \) is \( x^2 y^2 - 6xy + 9 \).

Step by step solution

01

Identify the terms of the binomial

Recognize that in the expression \( (xy - 3)^2 \), the term 'a' is 'xy' and the term 'b' is '3'.
02

Apply the formula for the square of a binomial

Using the formula for squaring a binomial \( (a - b)^2 = a^2 - 2ab + b^2 , replace 'a' with 'xy' and 'b' with '3': \( (xy - 3)^2 = (xy)^2 - 2*(xy)*3 + 3^2 \)
03

Simplify the expression

The result is \( x^2 y^2 - 6xy + 9 \).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions form the backbone of algebra. They consist of variables, numbers, and operations combined together. In the expression \( xy - 3 \), \( xy \) is a term where 'x' and 'y' are variables, and \(-3\) is a constant term. Algebraic expressions can take on various forms and can represent numbers or quantities depending on the values of the variables. To understand an algebraic expression, you should:
  • Identify each term in the expression
  • Recognize constants and coefficients
  • Understand how terms are combined using operations like addition and subtraction
This understanding helps in manipulating the expressions, such as expanding, factoring, or simplifying them.
Squaring Binomials
Squaring a binomial involves a specific algebraic formula you'll often encounter. A binomial is any algebraic expression with two terms, like \( (xy - 3) \). When you square a binomial, you're applying the formula:\[ (a - b)^2 = a^2 - 2ab + b^2 \]In this formula, 'a' and 'b' are the two terms of the binomial. Squaring \( xy - 3 \) involves:
  • Calculating \( (xy)^2 \), which is \( x^2y^2 \)
  • Performing \(-2ab\), which for this binomial is \(-2(xy)(3) = -6xy\)
  • Computing \( 3^2 \), resulting in \( 9 \)
These steps yield the expanded result: \( x^2y^2 - 6xy + 9 \). This consistent method helps in systematically handling more complex binomials.
Polynomial Expansion
Polynomial expansion refers to expressing a polynomial raised to a power as a sum of simpler terms. It involves applying the binomial theorem or using standard algebraic formulas. When expanding a polynomial, each term of the binomial is considered in combination with others. For squaring binomials, like \((xy - 3)^2\), you use its specific expansion formula, resulting in a polynomial:
  • Start with the highest degree term, \(a^2\).
  • Progress through middle term combinations, using \(-2ab\).
  • Ensure all terms are calculated and combined.
Polynomials can then be rearranged or simplified to facilitate understanding or further calculations. This process is crucial in simplifying complex algebraic equations and finding practical solutions in mathematics.

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Most popular questions from this chapter

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What polynomial must be subtracted from \(5 x^{2}-2 x+1\) so that the difference is \(8 x^{2}-x+3 ?\)

The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for \(2^{-1}+2^{-2}\) of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.

Use a graphing utility to graph each side of the equation in the same viewing rectangle. (Call the left side \(y_{1}\) and the right side \(y_{2} .\) I If the graphs coincide, verify that the multiplication has been performed correctly. If the graphs do not appear to coincide, this indicates that the multiplication is incorrect. In these exercises, correct the right side of the equation. Then graph the left side and the corrected right side to verify that the graphs coincide. \((x-2)(x+2)+4=x^{2} ;\) Use a \([-6,5,1]\) by \([-2,18,1]\) viewing rectangle.

Use a graphing utility to graph each side of the equation in the same viewing rectangle. (Call the left side \(y_{1}\) and the right side \(y_{2} .\) I If the graphs coincide, verify that the multiplication has been performed correctly. If the graphs do not appear to coincide, this indicates that the multiplication is incorrect. In these exercises, correct the right side of the equation. Then graph the left side and the corrected right side to verify that the graphs coincide. \((x+2)^{2}=x^{2}+2 x+4 ;\) Use a \([-6,5,1]\) by \([0,20,1]\) viewing rectangle.

Explain how to simplify an expression that involves a quotient raised to a power. Provide an example with your explanation.
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