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Chapter 4: Problem 2

Explain in your own words how to square a binomial. Give an example.

Short Answer

Expert verified
The square of a binomial \((a + b)^{2}\) is expanded using the formula \(a^{2} + 2ab + b^{2}\). For example, \((2 + 3)^{2} = 25\).

Step by step solution

01

- Write Down the Binomial

Start by identifying and writing down the binomial that you need to square. For example, let's take the binomial \((a + b)\).
02

- Apply the Binomial Squaring Formula

Use the formula for squaring a binomial: \((a + b)^{2} = a^{2} + 2ab + b^{2}\). This formula tells you how to expand the square of a binomial.
03

- Expand the Binomial

Expand the binomial using the formula. For \((a + b)^{2}\), apply the formula: \(a^{2} + 2ab + b^{2}\).
04

- Simplify the Expression

If needed, simplify the terms obtained after expansion. Since this example already appears simplified, no further steps are necessary for simplification.
05

- Use a Specific Example

Now use a specific example, such as \((2 + 3)^{2}\). Substitute \(a = 2\) and \(b = 3\) into the formula. According to the binomial squaring formula, \((2 + 3)^{2}\) becomes \((2^{2}) + 2(2)(3) + (3^{2}) = 4 + 12 + 9 = 25\).

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Key Concepts

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

Binomial Expansion
When you expand a binomial, you are multiplying it by itself or raising it to a power. In simpler terms, binomial expansion involves distributing or spreading the values within the binomial across each term.

For example, if you have \((a + b)^2\), you distribute each term in \((a + b)\) by using the formula for squaring a binomial. This process is known as binomial expansion.

To perform binomial expansion correctly:
  • Identify each term within the binomial.
  • Use the squaring formula or appropriate binomial theorem for higher powers.
The binomial squaring formula simplifies this process significantly, as explained in the next section.
Binomial Squaring Formula
The binomial squaring formula is a shortcut that helps you quickly square a binomial. This formula states: \((a + b)^2 = a^2 + 2ab + b^2\).

This formula means that to square a binomial:
  • Square the first term (i.e., \[a^2\])
  • Multiply the first and second terms together, then double it (i.e., \[2ab\])
  • Square the second term (i.e., \[b^2\])


For example, if we take \((2 + 3)^2\), using the formula,
\[ (2 + 3)^2 = 2^2 + 2(2)(3) + 3^2 = 4 + 12 + 9 = 25\] By using the binomial squaring formula, you can simplify the multiplication process and reduce the possibility of making errors.
Simplification of Algebraic Expressions
After expanding a binomial, the next step is simplification. Simplification involves combining like terms and resolving any arithmetic operations to reach the simplest form of the expression.

In our example, if you start with \((2 + 3)^2\):
  • First, apply the formula: \[2^2 + 2(2)(3) + 3^2\]
  • Perform the arithmetic operations inside the formula: \[4 + 12 + 9\]
  • Then, add the results together: \[25\]


Simplifying algebraic expressions ensures that the result is in its most concise form, making it easier to understand and use. It helps in reducing complexity and makes further calculations more manageable.

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Most popular questions from this chapter

Subtract. See Example \(\boldsymbol{\delta}\) $$ \begin{array}{l} {12 m^{3}-8 m^{2}+6 m+7} \\ {-3 m^{3}+5 m^{2}-2 m-4} \\ \hline \end{array} $$

Find each product. $$ \left(2 x^{2}-5\right)\left(2 x^{2}+5\right) $$

Find each product. $$ (r+5)^{3} $$

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