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Chapter 5: Problem 52

Multiply using the rules for the square of a binomial. $$(4 y-3)^{2}$$

Short Answer

Expert verified
The expanded form of \( (4y - 3)^{2} \) is \( 16y^{2} - 24y + 9 \).

Step by step solution

01

Identify the Elements

In the expression \( (4y - 3)^{2} \), the elements are \( a = 4y \) and \( b = 3 \).
02

Apply the Binomial Square Rule

Substitute \( a = 4y \) and \( b = 3 \) into the formula \( a^{2} - 2ab + b^{2} \). This gives us \( (4y)^{2} - 2(4y)(3) + (3)^{2} \).
03

Expand and Simplify

Next, calculate each component of the expression. \( (4y)^{2} = 16y^{2} \), \( 2(4y)(3) = 24y \), and \( (3)^{2} = 9. Combine these to get the final expression: \( 16y^{2} - 24y + 9 \).

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Key Concepts

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

Square of a Binomial
The square of a binomial is an essential concept in algebra that helps us to expand expressions like \((4y - 3)^2\). This involves utilizing a specific formula, which simplifies the process and avoids lengthy multiplication. The formula for the square of a binomial \((a - b)^2\) is expressed as:
  • \(a^2 - 2ab + b^2\)
To apply this to our example, first identify \(a\) and \(b\), which are \(4y\) and \(3\) respectively. Then substitute these into the formula to make the expansion straightforward. This method saves time and reduces errors since it uses directly derived components from the binomial itself.
Algebraic Expressions
An algebraic expression consists of variables, numbers, and operations (like addition, subtraction, etc.). In our example, \((4y - 3)^2\), the term \(4y - 3\) is an algebraic expression, where \(4y\) is a product of a coefficient (4) and a variable (\(y\)), and \(-3\) is a constant term.
  • Variables represent unknowns or can be placeholders for numbers.
  • Coefficients are the numerical parts that multiply the variables.
  • Constants are numbers on their own with no variables attached.
Understanding the structure of algebraic expressions is crucial, as they are the foundation for operations like expansion and simplification. Recognizing parts of the expression helps in correctly applying algebraic rules.
Expansion and Simplification
The process of expansion and simplification in algebra involves opening up expressions and reducing them to their simplest form. For example, for \((4y - 3)^2\), begin by expanding the expression using the binomial square formula described earlier.
  • Expand: \((4y)^2 - 2(4y)(3) + 3^2\)
  • Simplify: Compute each part: \(16y^2 - 24y + 9\)
Expansion gives us a longer expression and shows all interactions between the terms, while simplification reduces it to a streamlined form. This process brings clarity and is helpful in solving equations and evaluating expressions.
College Algebra
College algebra extends foundational algebraic concepts to more advanced topics, often encountered in various academic and professional fields. In the context of squaring binomials like \((4y - 3)^2\), students learn not only about expansion and simplification rules but also how these techniques apply in broader scenarios.
  • Advanced applications include solving polynomial equations.
  • Leverages knowledge in calculus, statistics, and other areas.
  • Important for logical reasoning and problem-solving skills.
Mastering concepts such as binomial expansion in college algebra provides the tools needed for higher-level mathematics and practical problem-solving in many disciplines.

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Most popular questions from this chapter

Explain how to subtract polynomials.

Use the second step to combine the like terms. $$5 x y+6 x y=(5+6) x y=?$$

Divide each expression using the quotient rule. Express any numerical answers in exponential form. $$\frac{x^{200} y^{40}}{x^{25} y^{10}}$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l}3 x+4 y=7 \\\2 x+7 y=9\end{array}\right.$$

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. As long as I understand how to add and subtract polynomials, I can select the format, horizontal or vertical, that works best for me.
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