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Chapter 0: Problem 12

The expanded form of the square of a binomial is a trinomial called a _____ square trinomial.

Short Answer

Expert verified
The expanded form of the square of a binomial is called a square trinomial.

Step by step solution

01

Understanding the Binomial Square

To solve this problem, start by understanding what a binomial square is. A binomial is an algebraic expression that contains two terms. The square of a binomial means multiplying the binomial by itself.
02

Expanding the Binomial Square

When you square a binomial, you apply the distributive property. For a binomial \( (a + b) \), the square of the binomial is calculated as follows: \[ (a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2 \]
03

Identifying the Trinomial

The result of expanding a binomial square is a trinomial. In this case, \[ a^2 + 2ab + b^2 \] is a trinomial, which has three terms.
04

Naming the Trinomial

The expanded form of the square of a binomial is called a square trinomial because it is derived by squaring a binomial.

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Key Concepts

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

Binomial
A binomial is an algebraic expression that consists of exactly two terms. These terms are usually joined by a plus (+) or minus (-) sign.
For example:
  • (a + b)
  • (x - y)
The key feature of a binomial is the presence of these two distinct terms.
When we talk about squaring a binomial, it means multiplying the binomial by itself. This often forms the basis for many algebraic operations and identities.
Trinomial
A trinomial is an algebraic expression containing exactly three terms. These terms can be any combination of coefficients and variables. For example:
  • (a^2 + 2ab + b^2)
  • (x^2 - 3x + 9)
In the context of squaring a binomial, the resulting expression is a trinomial. For instance, when we expand (a + b)^2, we get the trinomial a^2 + 2ab + b^2.
Here, a multinomial is an algebraic expression with two or more terms. So, both binomials and trinomials are specific types of multinomials.
This highlights the simplicity and power of multiplying binomials.
Distributive Property
The distributive property is a foundational property of arithmetic and algebra. It states that for all real numbers a, b, and c: a(b + c) = ab + ac
In the context of squaring a binomial, we use the distributive property to expand the expression.
For example:
  • (a + b)^2 = (a + b)(a + b)
  • Applying the distributive property: a(a + b) + b(a + b)
  • Continuing, we distribute each term: a^2 + ab + ba + b^2
  • Simplifying, we get a^2 + 2ab + b^2
This shows how the distributive property helps in expanding and simplifying binomials.
Algebraic Expression
An algebraic expression is a combination of variables, constants, and arithmetic operations (addition, subtraction, multiplication, division). For example:
  • x + 3
  • 4y^2 - 2y + 7
  • (a + b)^2
Algebraic expressions can represent anything from simple sums to complex functions. A binomial is a type of algebraic expression with two terms. A trinomial has three terms. Understanding algebraic expressions is crucial in algebra since they form the foundation for equations, inequalities, and functions.
By mastering simple expressions like binomials and trinomials, students can tackle more complex problems with confidence.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2005 .\)

Multiply and simplify. $$ 7 m^{2}\left(2 m^{4}-3 m+4\right) $$

Determine if the statement is true or false. The product of two polynomials each of degree 4 will be degree 8 .

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 5 \sqrt{7}(3 \sqrt{7}-2 \sqrt{5}+3) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (5 \sqrt{z}-4)(3 \sqrt{z}+6) $$
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