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Chapter 5: Problem 75

Multiply. See Examples I through II. $$ [(x y+4)-6]^{2} $$

Short Answer

Expert verified
The expanded form of \([(x y + 4) - 6]^{2}\) is \(x^{2}y^{2} - 4xy + 4\).

Step by step solution

01

Simplify the Expression Inside the Brackets

Start by simplifying the expression inside the brackets: \((x y + 4) - 6\). To do this, you subtract 6 from the constant term 4, resulting in the simplified expression \(x y - 2\).
02

Expand the Squared Expression

Expand the expression \((x y - 2)^{2}\). According to the formula \((a - b)^{2} = a^{2} - 2ab + b^{2}\), we substitute \(a = x y\) and \(b = 2\). Thus, \((x y - 2)^{2} = (x y)^{2} - 2(x y)(2) + (2)^{2}\).
03

Calculate Each Term

Calculate each term:1. \((x y)^{2} = x^{2}y^{2}\),2. \(-2(x y)(2) = -4xy\),3. \((2)^{2} = 4\).
04

Combine the Terms

Combine the calculated terms to form the final expanded expression: \(x^{2}y^{2} - 4xy + 4\).

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Key Concepts

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

Algebraic Expressions
In algebra, we often deal with expressions that include variables, numbers, and operators all combined together. These are known as algebraic expressions. For instance, in our problem, the expression \((xy + 4) - 6\) is an algebraic expression. It contains:
  • Variables: 'x' and 'y'
  • Constants: Numbers like 4 and -6
  • Operators: Addition and subtraction
Algebraic expressions can be simplified by combining like terms or using arithmetic operations. In our problem, we simplify \((xy + 4) - 6\) to \(xy - 2\) by performing the subtraction between the constants. Simplifying expressions makes them easier to work with, especially when you need to perform further operations like multiplication or expansion.
Binomial Expansion
The binomial expansion involves expressing a binomial raised to a power in an expanded form. A binomial is simply an algebraic expression containing two terms, like \((xy - 2)\). When we square a binomial, like we do in our problem, it follows the pattern of the binomial formula: \((a - b)^2 = a^2 - 2ab + b^2\). This is essentially expanding the binomial into simpler components to make calculations easier:
  • The first term \(a^2\) refers to squaring the first term of the binomial.
  • The middle term \(-2ab\) leverages both terms with multiplication and a factor of 2.
  • The last term \(b^2\) is simply the square of the second term.
By substituting \(a = xy\) and \(b = 2\), our binomial \((xy - 2)^2\) can be broken down into parts, each of which is calculated separately, simplifying our task of expansion.
Polynomials
Polynomials are algebraic expressions that include sums of powers of variables with coefficients. They can have one or more terms, which means every binomial and its expansions can be regarded as polynomials. For example, when \((xy - 2)^2\) is expanded, we get a polynomial: \(x^2y^2 - 4xy + 4\). Understanding polynomials involves:
  • Recognizing terms: Each term could be a single variable raised to a power, like \(x^2y^2\).
  • Coefficients: Numbers multiplying the variables, like -4 in \(-4xy\).
  • Degrees: The highest sum of the exponents in any term, which for \(x^2y^2\) is 4 (since \(x^1y^1\) has degrees of 2 each).
By learning how to handle polynomials, we can solve a variety of algebraic problems, whether we're multiplying, factoring, or finding roots.

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Most popular questions from this chapter

Simplify where possible. $$a. x^{a} \cdot x^{a}$$ $$b. x^{a}+x^{a}$$ $$c.\frac{x^{a}}{x^{b}}$$ $$d. x^{a} \cdot x^{b}$$ $$e. x^{a}+x^{b}$$

Factor. Assume that variables used as exponents represent positive integers. $$ 2 x^{2 n}+11 x^{n}+5 $$

Factor each trinomial. See Examples 5 through 10. $$ 3 x^{2}+14 x+15 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{4 n}-625 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 3 x^{2 n}-8 x^{n}+4 $$
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