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Chapter 5: Problem 46

Multiply using the rules for the square of a binomial. $$(x+5)^{2}$$

Short Answer

Expert verified
The square of the binomial \((x+5)^2\) is \(x^{2} + 10x + 25\).

Step by step solution

01

Identifying the terms in the binomial

The expression is \((x+5)^2\). Here, \(a\) is represented by \(x\) and \(b\) is represented by \(5\). This is because in the formula \(a^{2} + 2ab + b^{2}\), \(a\) and \(b\) represent the two terms in the formula.
02

Applying the rules for the square of a binomial

Substitute \(a\) and \(b\) into the formula \(a^{2} + 2ab + b^{2}\). Gaining \(x^{2} + 2*x*5 + 5^{2}\).
03

Solving further

Now it is a matter of analyzing and simplifying the numbers. \(x^{2} + 10x + 25\).

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Key Concepts

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

Algebra
Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It allows us to represent numbers in a general form using variables, and perform calculations that apply to many different numbers.

In algebra, expressions like \((x+5)^2\) become manageable. Here, \(x\) represents a variable, while \(5\) is a constant. Algebra helps us simplify and solve these expressions using specific formulas.

This exercise focuses on squaring a binomial using the rule \((a+b)^2 = a^2 + 2ab + b^2\). This formula simplifies the multiplication process, providing a straightforward method to expand squared binomials.
Binomial Multiplication
Binomial multiplication involves multiplying expressions that have two terms. When squaring a binomial like \((x+5)\), you're essentially multiplying \((x+5)\) by \((x+5)\).

The formula for squaring a binomial is \((a+b)^2 = a^2 + 2ab + b^2\). This clearly states:
  • Square the first term: \(a^2\)
  • Multiply the two terms together and double it: \(2ab\)
  • Square the second term: \(b^2\)
For \((x+5)\), this becomes:
  • \(x^2\)
  • \(2 \times x \times 5 = 10x\)
  • \(5^2 = 25\)
Putting it all together, the result is \(x^2 + 10x + 25\).

This approach simplifies what could be a tedious multiplication process, making it easier to manage.
Polynomial Expressions
Polynomial expressions are made up of terms that include variables raised to whole-number exponents. In our solution, \(x^2 + 10x + 25\) is a polynomial expression.

Each part of this polynomial has a significance:
  • \(x^2\) is the quadratic term, representing a squared variable.
  • \(10x\) is the linear term, showing the variable multiplied by a constant.
  • \(25\) is the constant term, representing a fixed number.
Polynomials like these are integral in algebra because they form the basis for more complex functions and equations.

The goal is to simplify these expressions, allowing for easier manipulation and understanding of algebraic concepts.

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Most popular questions from this chapter

What is a monomial? Give an example with your explanation.

What is the degree of a polynomial? Provide an example with your explanation.

Solve: \(8-6 x>4 x-12\). (Section \(2.7,\) Example 7 )

Will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{\left(x^{2}\right)^{3}}{5^{3}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(6 x^{2} y-7 x y-4\right)-\left(6 x^{2} y+7 x y-4\right)=0$$
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