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Chapter 5: Problem 219

Square each binomial using the Binomial Squares Pattern. $$ (2 y-3 z)^{2} $$

Short Answer

Expert verified
The result is \ 4y^2 - 12yz + 9z^2 \

Step by step solution

01

Identify the Binomial

Recognize the binomial expression \( (2y - 3z) \). It consists of two parts: \ 2y \ and \ -3z \.
02

Recall the Binomial Squares Pattern

Remember the Binomial Squares Pattern: \( (a - b)^2 = a^2 - 2ab + b^2 \) where \( a = 2y \) and \( b = 3z \).
03

Square the First Term

Square the first term, \( (2y)^2 \). \[ (2y)^2 = 4y^2 \]
04

Multiply the Two Terms and Double It

Multiply the two terms \( 2y \ and \ -3z \), then double the product: \[ 2 \times (2y \times -3z) = -12yz \]
05

Square the Second Term

Square the second term, \( (-3z)^2 \). \[ (-3z)^2 = 9z^2 \]
06

Combine the Results

Add the results together: \[ 4y^2 - 12yz + 9z^2 \]

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Key Concepts

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

Binomial Expression
A binomial expression is a polynomial with two terms. The expression consists of two variables or constants connected by either addition or subtraction. For example, in the given exercise, the binomial is \((2y - 3z)\). Here are some key takeaways to understand binomials better:

  • Binomials have precisely two terms.
  • The terms can include variables, coefficients, or both.
  • Examples: \(a + b\), \(5x - 4\), and \(3p + 7q\).
Understanding the structure of a binomial is critical as it allows you to apply various mathematical patterns and formulas accurately.
Squaring Binomials
Squaring a binomial means multiplying the binomial by itself. This process follows a specific pattern known as the Binomial Squares Pattern. The formula for this pattern is:

\[ (a \,\text{卤}\, b)^2 = a^2 \, \text{卤} \, 2ab + b^2 \]
For example, with the binomial \((2y - 3z)\):
  • First, square the first term: \(a = 2y\), so \(a^2 = (2y)^2 = 4y^2\).
  • Second, multiply both terms and double the result: \(2 \times (2y \times -3z) = -12yz\).
  • Third, square the second term: \(b = -3z\), so \(b^2 = (-3z)^2 = 9z^2\).
  • Finally, combine the results: \(4y^2 - 12yz + 9z^2\).
Simplifying binomials using this pattern makes solving expressions much easier.
Polynomial Expansion
Polynomial expansion involves expressing a polynomial in an extended form. Squaring binomials is a special case of polynomial expansion, because the result is a trinomial. Here are steps to follow in polynomial expansion:
1. Identify the binomial and recognize its terms.
2. Apply the Binomial Squares Pattern, which simplifies the process.
3. Expand the polynomial by carefully following the steps:
  • Square the first term, \(a^2\).
  • Multiply the two terms and double the product, \(2ab\).
  • Square the second term, \(b^2\).
  • Combine all results, ensuring proper signs.
In our example of \((2y - 3z)^2\), the expanded form is \(4y^2 - 12yz + 9z^2\). Polynomial expansion allows us to see how each term contributes to the final expression.

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