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Chapter 10: Problem 15

Identify the coefficient of each term. a. \(-5 x^{2} y\) b. \(12 a^{5}\) c. 6 d. \(w z\)

Short Answer

Expert verified
a. -5, b. 12, c. 6, d. 1

Step by step solution

01

Identify the coefficient in \(-5 x^{2} y\)

The coefficient is the numerical factor that multiplies the variables in a term. In the term \(-5 x^{2} y\), the coefficient is -5.
02

Identify the coefficient in \(12 a^{5}\)

The coefficient in the term \(12 a^{5}\) is the number 12, which multiplies the variable \(a^{5}\).
03

Identify the coefficient in 6

The term 6 does not have any variables, so the coefficient is simply the number 6 itself.
04

Identify the coefficient in \(w z\)

When variables appear without a numerical factor, the coefficient is understood to be 1. Therefore, the coefficient in the term \(w z\) is 1.

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Key Concepts

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

Identifying Coefficients
In algebra, coefficients are the numerical parts of terms that multiply the variables. When looking at algebraic expressions, coefficients can be positive, negative, whole numbers, or even fractions. To find the coefficient, you need to identify the number that comes before the variables. For example, in the term \(-5 x^{2} y\), the coefficient is \(-5\). This means that \(-5\) is multiplying the variables \((x^{2} y)\). It is essential to not confuse the coefficient with the variables themselves.
Numerical Factor
The numerical factor in a term is another name for the coefficient. It's the number that is multiplied by the variables within a term. It doesn't matter how many variables there are; the numerical factor remains the number part of the term. For instance, in \(12 a^{5}\), the numerical factor is \(12\). Even if the variables change, the numerical factor remains the coefficient. Identifying the numerical factor is key because it shows how the term scales with respect to its variables.
Algebraic Terms
An algebraic term consists of a coefficient, variables, and their exponents. Terms are usually separated by a plus or minus sign in an equation or expression. They can be as simple as a single number, such as \6\, or more complex like \(-5 x^{2} y\).
  • The coefficient is the numerical part.
  • The variables are the letters that represent numbers.
  • The exponents show how many times a variable is multiplied by itself.
For example, in the term \(6abc^{2}\), \(6\) is the coefficient, \(a\), \(b\), and \(c\) are the variables, and \(c^{2}\) indicates that \(c\) is squared.
Variable Multiplication
Variable multiplication involves combining variables and coefficients. When multiple variables are multiplied together, their coefficients are also multiplied. For instance, when you see \(w z\), it means \(w\) and \(z\) are multiplied together. If a term involves variables without a visible numerical factor, the coefficient is understood to be \1\. In \(w z\), the coefficient is \1\ because \(w z = 1 \times w \times z\). Understanding how to multiply variables and combine coefficients is crucial in simplifying and solving algebraic expressions.

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