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Chapter 1: Problem 40

Perform the indicated multiplications. $$(x-3 y)^{2}$$

Short Answer

Expert verified
\( (x - 3y)^2 = x^2 - 6xy + 9y^2 \)

Step by step solution

01

Identify the Binomial

Recognize that the expression \((x - 3y)^2\) is a binomial squared. It consists of two terms: \(x\) and \(-3y\).
02

Apply the Binomial Square Formula

Use the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 3y\).
03

Calculate Each Term

Substitute \(a = x\) and \(b = 3y\) into the formula:- First term: \(a^2 = x^2\).- Second term: \(-2ab = -2(x)(3y) = -6xy\).- Third term: \(b^2 = (3y)^2 = 9y^2\).
04

Combine All Terms

Combine the calculated terms to express the squared binomial:\[(x - 3y)^2 = x^2 - 6xy + 9y^2\].

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a pivotal concept in algebra that allows us to expand expressions that are raised to a power. When we see a binomial expression like \((x - 3y)^2\), we know it's an expression with two terms.
The theorem provides a formula to expand the expression and find all its terms within seconds, rather than manually multiplying it out. The square of a binomial follows this particular pattern:
  • \((a - b)^2 = a^2 - 2ab + b^2\)
It's all about recognizing the components and applying the formula correctly. In our example, the binomial theorem simplifies finding the expansion without cumbersome calculations. Once you understand this pattern, you can quickly determine the result of any similar expression.
Polynomial Expansion
Polynomial expansion refers to the process of expressing a polynomial in its simplest form by multiplying out all factors. For a binomial like \((x - 3y)^2\), the goal is to expand it into an expression only involving separate terms.
Here's how we achieve this with our binomial:
  • Identify the formula: \((a - b)^2\)
  • Substitute the variables: \(a = x\) and \(b = 3y\)
  • Calculate each part: \(x^2\), \(-6xy\), and \(9y^2\)
By following these straightforward steps, we obtain the expanded expression: \[x^2 - 6xy + 9y^2\].
Understanding polynomial expansion is crucial for algebra, as it lays the groundwork for simplifying and analyzing more complex polynomial expressions.
Mathematical Expressions
Mathematical expressions are the building blocks of algebra. They involve numbers, variables, and operations to create meaningful equations or functions. When working with mathematical expressions like \((x - 3y)^2\), it's important to recognize their components and the operations involved.
A mathematical expression can be:
  • Simple, like \(a + b\)
  • Complex, involving powers like \((x - 3y)^2\)
Expressions can be manipulated using rules and formulas, like the Binomial Theorem, to reveal underlying patterns or simplified forms. Mastering mathematical expressions enhances the ability to solve real-world problems and is a foundational skill in mathematics, essential for any serious study of algebra.

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Most popular questions from this chapter

Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. Two gasoline distributors, A and B, are 228 mi apart on Interstate \(80 .\) A charges \(\$ 3.40 /\) gal and \(\mathrm{B}\) charges \(\$ 3.20 /\) gal. Each charges \(0.2 \mathrm{g} / \mathrm{gal}\) per mile for delivery. Where on Interstate 80 is the cost to the customer the same?

Solve the given equations. All numbers are approximate. $$\frac{x}{2.0}=\frac{17}{6.0}$$

Solve the given equations. $$|7-x|=1$$

Solve the given problems. To find the amount of a certain investment of \(x\) dollars, it is necessary to solve the equation \(0.03 x+0.06(2000-x)=96\) Solve for \(x\).

Solve the given problems. In finding the maximum operating temperature \(T\) (in \(^{\circ} \mathrm{C}\) ) for a computer integrated circuit, the equation \(1.1=(T-76) / 40\) is used. Find the temperature.
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