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Chapter 5: Problem 216

Square each binomial using the Binomial Squares Pattern. $$ (w+4)^{2} $$

Short Answer

Expert verified
The expanded form is \( w^2 + 8w + 16 \).

Step by step solution

01

Understand the Binomial Squares Pattern

The binomial squares pattern is given by \( (a+b)^2 = a^2 + 2ab + b^2 \). This pattern helps in expanding the square of a binomial quickly.
02

Identify the Components

In the binomial \( (w+4) \), identify \( a \) as \( w \) and \( b \) as \( 4 \).
03

Apply the Binomial Squares Pattern

Substitute \( w \) for \( a \) and \( 4 \) for \( b \) into the pattern: \( (w+4)^2 = w^2 + 2 \cdot w \cdot 4 + 4^2 \).
04

Simplify the Expression

Perform the arithmetic operations inside the expansion: \( w^2 + 8w + 16 \).

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Key Concepts

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

Expanding Binomials
Expanding binomials is an essential skill in algebra. A binomial is an algebraic expression consisting of two terms, such as \( (w + 4) \). When we square a binomial, we use the binomial squares pattern, which is \( (a+b)^2 = a^2 + 2ab + b^2 \).
This pattern helps simplify the process and ensures accuracy.
To expand a binomial like \( (w + 4) \), follow these steps:
  • First, identify the components where \( a = w \) and \( b = 4 \).
  • Next, apply the pattern by substituting \( w \) for \( a \) and \( 4 \) for \( b \): \( (w + 4)^2 = w^2 + 2 \cdot w \cdot 4 + 4^2 \).
  • Finally, simplify the expression by performing the arithmetic operations: \( w^2 + 8w + 16 \).
By following these steps, you can efficiently expand any binomial.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can consist of numbers, variables, and operation symbols.
It does not have an equality sign like an equation.
Expanding binomials is part of manipulating algebraic expressions.
When working with algebraic expressions:
  • Identify each term in the expression. In \( (w + 4) \), the terms are \( w \) and \( 4 \).
  • Understand that variables like \( w \) represent unknown values.
  • Use algebraic rules and patterns, such as the binomial squares pattern, to simplify and solve expressions.
Manipulating algebraic expressions is foundational for more advanced algebra and calculus.
Mastery of these concepts simplifies both solving and understanding complex mathematical problems.
Polynomial Operations
Polynomial operations include addition, subtraction, multiplication, and division of polynomials.
Expanding binomials is a form of polynomial multiplication.
Polynomials are expressions with multiple terms, and they can include variables raised to different powers.
For example, in the expression \( w^2 + 8w + 16 \) expanded from \( (w + 4)^2 \):
  • \( w^2 \) is the quadratic term.
  • \( 8w \) is the linear term.
  • \( 16 \) is the constant term.
Polynomial operations also involve:
  • Combining like terms to simplify expressions.
  • Using the distributive property, such as \( a(b + c) = ab + ac \), to expand products of polynomials.
Understanding polynomial operations allows you to simplify, solve, and factor complex algebraic expressions effectively.

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Most popular questions from this chapter

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (9 c+5)(9 c-5) $$

Multiply. Use either method. $$ (3 q+1)\left(q^{2}-4 q-5\right) $$

Multiply the binomials. Use any method. $$ (2 t-9)(10 t+1) $$

Find each product. $$ \left(r^{2}-s^{3}\right)\left(r^{3}+s^{2}\right) $$

Find each product. $$ \left(m^{3}-8 n\right)^{2} $$
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