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Chapter 0: Problem 10

Complete each statement. The numerical coefficient in the term \(-7 y z^{2}\) is _____.

Short Answer

Expert verified
-7

Step by step solution

01

- Identify the Term

The given term is \(-7 y z^{2} \).
02

- Understand the Components

This term has three components: \(-7 \), which is a number, called the coefficient, and \(y z^{2} \), which are the variables.
03

- Extract the Coefficient

The numerical coefficient is the number that multiplies the variables. It is \(-7\) in this case.

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Key Concepts

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

Numerical Coefficient
In algebra, understanding the concept of a numerical coefficient is crucial.

Numerical coefficients are the numbers that sit in front of variables in an algebraic term.

Take the term \(-7 y z^2\) as an example. Here, the number \-7\ is being multiplied by the variables \(y\) and \(z^2\). Hence, \-7\ is the numerical coefficient of this term.

Algebraic expressions can have different types of numerical coefficients.

  • Positive Coefficient: For instance, in the term \(+5x^2\), the coefficient is 5.
  • Negative Coefficient: In the term \(-7y z^2\), the coefficient is -7.
  • Fractional Coefficient: Like in \(\frac{1}{2}x\), where \(\frac{1}{2}\) is the coefficient.
  • Decimal Coefficient: As in \(-0.3a b\), with -0.3 being the coefficient.

Understanding this concept helps you simplify and solve algebraic expressions and equations.
Variables in Algebra
Variables are symbols used to represent unknown values in algebraic expressions.

Typically, variables are denoted by letters such as \(x, y, z\), but any symbol can serve as a variable.

In the term \(-7 y z^2\), \(y\) and \(z^2\) are the variables.

Here are some key points about variables:
  • They are placeholders for numbers we don't know yet or that can change.
  • They allow us to create general solutions that work for a range of numbers.
  • They can be raised to exponents, like \((z^2)\), representing powers or repeated multiplication.

The power of variables is that they make it possible to write general mathematical rules and formulas that apply in many situations, not just specific instances.
Terms in Algebra
A term in algebra is a single mathematical expression that can be a constant, a variable, or the product of constants and variables.

The term \(-7 y z^2\) is made up of one numerical coefficient \(-7\) and two variables \(y\) and \(z^2\).

Key characteristics of algebraic terms include:
  • A term can be a standalone number like \(5\) or \(-3\).
  • It can also be just a variable, for example, \(x\) or \(z^2\).
  • Most commonly, it is a combination of coefficients and variables, like \(4x\) or \(-3a^2b\).
  • Terms are usually separated by addition or subtraction operations in expressions.

Recognizing and understanding terms in algebra helps in performing operations like simplifying expressions, solving equations, and factoring polynomials.

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Most popular questions from this chapter

Use an inequality symbol to write each statement. $$ 5+0 \text { is not equal to } 0 $$

Complete each statement so that the indicated property is illustrated. Simplify each answer if possible. \(0(0.35 t+12 u) =\frac{ }{\text { (multiplication property of 0 )}}\)

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ 10 \geq 10 $$

When simplifying an expression, we usually do some steps mentally.Providing the property that justifies each statement in the given simplification. $$3 x+4+2 x+7$$ $$ =3 x+(2 x+4)+7 $$

Simplify each expression. $$ -14\left(-\frac{2}{7}\right) \div(2 \cdot 6-10) $$
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